SPH3U Physics Fall 2024 PDF

PHYSICS FALL 2024 ARASH TASH 1. Kinematics - Motion in a straight line - Motion in 2 dimensions - Concepts include: displacement, velocity, acceleration, graphical analysis, projectile motion 2. Forces - Newton's laws of motion - Applications of f...

PHYSICS FALL 2024 ARASH TASH 1. Kinematics - Motion in a straight line - Motion in 2 dimensions - Concepts include: displacement, velocity, acceleration, graphical analysis, projectile motion 2. Forces - Newton's laws of motion - Applications of forces - Concepts include: types of forces, free-body diagrams, friction, gravity 3. Energy and Society - Work, energy, power - Thermal energy and society - Nuclear energy and society - Concepts include: energy transformations, conservation of energy, efficiency, heat 4. Waves and Sound - Vibrations and waves - Wave interactions - Concepts include: properties of waves, resonance, interference, Doppler effect 5. Electricity and Magnetism - Electricity and its production - Electromagnetism - Concepts include: electric circuits, magnetic fields, electromagnetic induction  Each unit typically has two assessment components: quiz and test 70% The course includes a final cumulative lab investigation and independent study project (ISP) worth 10%. There is a final exam worth 20% of the overall grade. The total assessment components add up to 100% of the course grade. Everyday life: Understanding how appliances work Principles behind sports techniques Safe driving practices Technology: Smartphones: touchscreens, accelerometers Computers: semiconductor physics Medical devices: MRI, ultrasound, X-rays Careers: Engineering: mechanical, electrical, aerospace Medicine: radiology, medical physics Research: particle physics, astrophysics Space exploration: NASA, CSA, SpaceX WHAT IS KINEMATICS? Definition: The study of motion Key concepts : Position Distance and displacement Speed and velocity Acceleration Importance: Foundational for understanding forces and energy POSITION Definition: The location of an object relative to a reference point Measured in metres (m) in SI system Examples: A car is 50 m north of the intersection A bird is 10 m above the ground Importance of choosing a reference point DISTANCE VS. DISPLACEMENT Distance: Total path length traveled Scalar quantity (magnitude only) Always positive Displacement: Change in position (vector) Has magnitude and direction Can be positive, negative, or zero Example: Walking around a 400 m track Distance = 400 m Displacement = 0 m SPEED VS. VELOCITY Speed: How fast an object is moving Scalar quantity Units: m/s, km/h Velocity: Speed in a given direction Vector quantity Units: m/s [direction], km/h [direction] Example: Car traveling 60 km/h north Speed = 60 km/h Velocity = 60 km/h [N] AVERAGE SPEED AND AVERAGE VELOCITY Average Speed = Total Distance / Total Time Average Velocity = Displacement / Time Interval Examples: Driving 240 km in 3 hours: Average speed = 80 km/h Ending 50 km east after 2 hours: Average velocity = 25 km/h [E] MOTION GRAPHS Position-time graphs: Slope represents velocity Straight line: constant velocity Curved line: changing velocity Velocity-time graphs: Slope represents acceleration Area under curve represents displacement REAL-WORLD APPLICATIONS OF KINEMATICS Estimate speeds of: Walking Running (sprinting) Cycling Car on highway Commercial airplane Compare estimates to actual values Discuss importance of estimation in physics Sports: Analyzing athlete performance, optimizing techniques Transportation: Designing safer vehicles, traffic management Space Exploration: Planning satellite orbits, spacecraft trajectories  Next class: Uniform motion and detailed analysis of position-time graphs Homework :Read textbook sections 1.1 and 1.2 REVIEW OF PREVIOUS LESSON Position, distance, displacement Speed vs. velocity Average speed and average velocity INTRODUCTION TO UNIFORM MOTION Definition: Motion with constant velocity (speed and direction) Characteristics: Constant speed Straight-line path Zero acceleration Real-world examples: Car on cruise control on a straight highway Conveyor belt in a factory POSITION-TIME GRAPHS FOR UNIFORM MOTION Characteristics of the graph: Straight line (linear relationship between position and time) y-intercept represents initial position Slope represents velocity Interpreting the graph: Positive slope: Motion in positive direction Negative slope: Motion in negative direction Steeper slope: Higher speed Horizontal line: Object at rest (zero velocity) Example graph: Position-time graph for object moving at 5 m/s for 10 seconds How to read initial position, final position, and displacement Multiple objects: How to represent multiple objects on the same graph How to compare their motions visually  CALCULATING VELOCITY FROM POSITION -TIME GRAPHS Formula: Velocity = Change in position / Change in time On the graph: Velocity = Slope of the line Step-by-step calculation: 1. Choose two points on the line (x₁, y₁) and (x₂, y₂) 2. Calculate rise: Δy = y₂ - y₁ (change in position) 3. Calculate run: Δx = x₂ - x₁ (change in time) 4. Calculate slope: m = Δy / Δx Example: Given: Position-time graph of a car Point 1: (0 s, 0 m), Point 2: (5 s, 20 m) Velocity = (20 m - 0 m) / (5 s - 0 s) = 4 m/s Practice: Calculate velocity for different sections of a multi-segment position-time graph UNIFORM MOTION EQUATION Primary equation: d = vt (where d is displacement, v is velocity, t is time) Rearrangements: v = d/t (useful for finding velocity) t = d/v (useful for finding time) Units: Displacement (d): meters (m) Velocity (v): meters per second (m/s) Time (t): seconds (s) Importance of consistent units Example problem: A car travels at a constant velocity of 20 m/s for 30 seconds. What is its displacement? Solution: d = vt = (20 m/s)(30 s) = 600 m MULTIPLE REPRESENTATIONS OF UNIFORM MOTION Example: "A car travels at a constant speed of 60 km/h for 2 hours." Data table: Time (h) | Position (km) (0, 0) (1,60) (2,120) Position-time graph Velocity-time graph d = vt = 60 km/h * 2 h = 120 km Convert between representations for a given scenario of uniform motion UNIFORM MOTION IN OPPOSITE DIRECTIONS Positive and negative velocities: Positive: Motion in the direction defined as positive Negative: Motion in the opposite direction Effect on position-time graph: Positive velocity: Line slopes upward Negative velocity: Line slopes downward Example: Two cars on a straight road: Car A: Moving east at 30 m/s (positive direction) Car B: Moving west at 20 m/s (negative direction) Show position-time graphs for both cars on the same axes Intersection point: Explain how to find when and where the cars will meet INTRODUCTION TO NON-UNIFORM MOTION Definition: Motion where velocity changes (introduction to acceleration) Characteristics of position-time graph: Curved line (parabola for constant acceleration) Changing slope (instantaneous velocity varies) Examples: Car speeding up: Upward curving line Ball thrown vertically upward: Parabola Object slowing down: Downward curving line Importance: Most real-world motions are non-uniform Segway to next lesson on acceleration  Sample problems from Nelson Physics 11 textbook: Section 1.2: Questions 1, 3, 5 Section 1.4: Questions 1, 2 REAL-WORLD APPLICATIONS Transportation: Calculating travel times for long-distance trips Determining fuel efficiency at constant speed Planning train schedules Sports: Analyzing constant-speed portions of races (e.g., middle of a 400m run) Pacing strategies for long-distance events Manufacturing: Designing conveyor systems for assembly lines Calibrating robotic arms for repetitive tasks Space exploration: Calculating satellite orbits Determining spacecraft trajectories during non-propulsion phases Environmental science: Modeling constant wind speeds for wind energy calculations Studying ocean currents PREVIEW OF NEXT LESSON AND HOMEWORK Next class:Velocity-time graphs Introduction to acceleration Homework: Read textbook sections 1.3-1.4 Practice problems: Section 1.3: Questions 2, 4, 7 Section 1.4: Questions 3, 5, 8 (Include full problem statements) In what situations might the assumption of uniform motion be useful, even if the motion isn't perfectly uniform? SCIENTIFIC NOTATION Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form, since to do so would require writing out an inconveniently long string of digits. In scientific notation, nonzero numbers are written in the form: m × 10n E NOTATION Calculators and computer programs typically present very large or small numbers using scientific notation, and some can be configured to uniformly present all numbers that way. Because superscript exponents like 107 can be inconvenient to display or type, the letter "E" or "e" (for "exponent") is often used to represent "times ten raised to the power of", so that the notation m E n for a decimal significand m and integer exponent n means the same as m × 10n. For example 6.022×1023 is written as 6.022E23 or 6.022e23, and 1.6×10−35 is written as 1.6E-35 or 1.6e-35.  The exponent of ten is always selected to be divisible by three to match the common metric prefixes, i.e. scientific notation that aligns with powers of a thousand, for example, 531×103 instead of 5.31×105 (but on calculator displays written without the ×10 to save space). REVIEW OF PREVIOUS LESSON Uniform Motion: Definition: Motion with constant velocity (speed and direction) Equation: d = vt (displacement = velocity × time) Position-Time (p-t) Graphs: Straight line for uniform motion Slope represents velocity Quick Quiz: Show p-t graph for object moving at 5 m/s for 10 s What does a horizontal line on a p-t graph represent? How do you calculate velocity from a p-t graph? INTRODUCTION TO VELOCITY-TIME GRAPHS Definition: Graph showing velocity vs. time Axes: x-axis: time (usually in seconds) y-axis: velocity (usually in m/s) Characteristics for Uniform Motion: Horizontal line (constant velocity) y-intercept represents initial velocity Distance above/below x-axis shows magnitude of velocity Interpretation: Above x-axis: positive velocity (motion in positive direction) Below x-axis: negative velocity (motion in negative direction) On x-axis: zero velocity (object at rest) Example: v-t graph for car traveling at constant 20 m/s for 30 s ANALYZING VELOCITY-TIME GRAPHS Slope: Represents acceleration (change in velocity over time) Flat line: zero acceleration (uniform motion) Upward slope: positive acceleration Downward slope: negative acceleration Area Under the Curve: Represents displacement For rectangular area: displacement = velocity × time Examples (show graphs for each):Constant positive velocity: horizontal line above x-axis 1. Constant negative velocity: horizontal line below x-axis 2. Object at rest: horizontal line on x-axis 3. Object speeding up: upward sloping line 4. Object slowing down: downward sloping line How would you represent a car speeding up, then maintaining constant speed, then slowing down to a stop? CALCULATING DISPLACEMENT FROM VELOCITY- TIME GRAPHS Displacement = Area under the velocity-time curve For uniform motion (rectangular area): d = v × t Steps to calculate: Identify the shape under the curve Calculate the area of that shape Pay attention to units Example Calculations: 1. Constant velocity: 1. Given: v = 5 m/s for 10 s 2. Area = 5 m/s × 10 s = 50 m 2. Triangular area (accelerating from rest): 1. Given: v increases from 0 to 10 m/s over 5 s 2. Area = ½ × base × height = ½ × 5 s × 10 m/s = 25 m Practice: Calculate displacement for a v-t graph with multiple segments RELATIONSHIP BETWEEN POSITION- TIME AND VELOCITY-TIME GRAPHS Comparison: p-t graph: position vs. time v-t graph: velocity vs. time Connections: Slope of p-t graph at any point = y-value on v-t graph at same time Area under v-t graph = total displacement on p-t graph Example: Show p-t graph for object with constant velocity Show corresponding v-t graph Demonstrate how slope of p-t matches v-t graph Practice: Given a p-t graph with changing slope, sketch the corresponding v-t graph Discuss how changes in p-t slope reflect on v-t graph INTRODUCTION TO ACCELERATION Definition: Rate of change of velocity over time Formula: a = Δv / Δt = (v_f - v_i) / t a: acceleration Δv: change in velocity Δt: change in time v_f: final velocity v_i: initial velocity Units: meters per second squared (m/s²) Types of Acceleration: Positive acceleration: Speeding up in positive direction Slowing down in negative direction Negative acceleration: Slowing down in positive direction Speeding up in negative direction Real-world examples: Car accelerating from stop light Skydiver reaching terminal velocity Rocket launch Is it possible to have negative acceleration when speeding up? UNIFORM ACCELERATION Definition: Constant rate of change of velocityCharacteristics on v-t graph:Straight line (not horizontal) Slope represents acceleration Examples (show v-t graphs for each): Car speeding up: upward sloping line starting from zero 1. Object in free fall: downward sloping line (neglecting air resistance) 2. Car braking to a stop: downward sloping line ending at zero Equation: v_f = v_i + atv_f: final velocity v_i: initial velocity a: acceleration t: time Given initial velocity, acceleration, and time, calculate final velocity CALCULATING ACCELERATION FROM VELOCITY-TIME GRAPHS Formula: a = (v_f - v_i) / t Graphical interpretation: Slope of v-t graph Steps to calculate: Identify two points on the line (t1, v1) and (t2, v2) Calculate slope: (v2 - v1) / (t2 - t1) Example calculations: Positive acceleration: Given: v changes from 0 to 10 m/s in 5 s a = (10 m/s - 0 m/s) / 5 s = 2 m/s² Negative acceleration (deceleration): Given: v changes from 20 m/s to 5 m/s in 3 s a = (5 m/s - 20 m/s) / 3 s = -5 m/s² Calculate acceleration for various v-t graphs POSITION-TIME GRAPHS FOR ACCELERATED MOTION Characteristics: Curved line (parabola for uniform acceleration) Changing slope (instantaneous velocity varies) Comparison with uniform motion p-t graphs: Uniform motion: straight line Accelerated motion: curved line Examples (show graphs): Object accelerating from rest Object decelerating to a stop Object thrown upward and falling back down Instantaneous velocity: Tangent line to curve at any point Slope of tangent line gives velocity at that instant How would you determine the point of maximum height for an object thrown upward? REAL-WORLD APPLICATIONS OF ACCELERATION Transportation: Car acceleration and braking Highway on-ramp design Aircraft takeoff and landing Sports: Sprinter's start Long jump takeoff Diving board mechanics Amusement park rides: Roller coasters (positive and negative acceleration) Free-fall rides Space exploration: Rocket launches Re-entry deceleration Technology: Hard drive read/write heads Elevator motion Natural phenomena: Falling objects (with and without air resistance) Tectonic plate movement PREVIEW OF NEXT LESSON AND HOMEWORK Next class topics: More on acceleration Equations of motion for uniform acceleration Free fall motion Homework: Read textbook sections 1.5-1.6 Practice problems: Section 1.5: Questions 2, 4, 6 Section 1.6: Questions 3, 5, 7 Reflection question: "How might understanding acceleration help you in everyday life or in a future career?" REVIEW OF PREVIOUS LESSON Velocity-Time Graphs: Horizontal line: constant velocity Sloped line: accelerating/decelerating Area under curve: displacement Acceleration: Definition: rate of change of velocity Formula: a = Δv / Δt Units: m/s² Uniform Acceleration: Constant rate of change of velocity Linear v-t graph ACCELERATION IN EVERYDAY LIFE Positive Acceleration Examples: Car speeding up: 0 to 100 km/h in 8 seconds (a ≈ 3.5 m/s²) Rocket launch: 0 to 1600 km/h in 60 seconds (a ≈ 7.4 m/s²) Falling object: increases speed by 9.8 m/s every second Negative Acceleration (Deceleration) Examples: Car braking: 100 km/h to 0 in 4 seconds (a ≈ -6.9 m/s²) Ball thrown upwards: slows down by 9.8 m/s every second Parachutist reaching terminal velocity: from 200 km/h to 20 km/h Discussion: How does acceleration feel in your body? Why is understanding acceleration important for vehicle safety? AVERAGE Average Acceleration: ACCELERATION Formula: a_avg = Δv / Δt = (v_f - v_i) / (t_f - t_i) Represents overall change in velocity over a time VS. interval INSTANTANEOUS Example: Car accelerating from 0 to 60 km/h in 10 ACCELERATION seconds a_avg = (60 km/h - 0) / 10 s = 6 km/h/s = 1.67 m/s² Instantaneous Acceleration: Acceleration at a specific moment in time Represented by the slope of the tangent line on a v-t graph Example: Acceleration of a bouncing ball at the moment it hits the ground Graphical Representation: v-t graph with changing acceleration Find average and instantaneous acceleration ACCELERATION DUE TO GRAVITY Definition: Acceleration experienced by objects in free fall near Earth's surface Symbol: g Standard Value: 9.8 m/s² downward (approximate) Variations: At sea level: 9.832 m/s² at poles, 9.780 m/s² at equator Decreases with altitude: 9.764 m/s² at 1000 m above sea level Importance: Fundamental constant in physics calculations Used in projectile motion, energy calculations, orbital mechanics Interesting Facts: Moon's gravity: about 1.62 m/s² Jupiter's gravity: about 24.79 m/s² How would different gravitational accelerations affect human exploration of other planets? EQUATIONS OF MOTION FOR UNIFORM ACCELERATION Assumptions: Constant acceleration Motion in a straight line The Four Main Equations: v = v₀ + at Δx = ½(v + v₀)t Δx = v₀t + ½at² v² = v₀² + 2aΔx Variable Definitions: v: final velocity v₀: initial velocity a: acceleration t: time interval Δx: displacement Importance: These equations allow us to solve a wide range of motion problems Form the foundation for more complex motion analysis DERIVING THE EQUATIONS OF MOTION Basic Definitions: a = Δv / Δt v_avg = Δx / Δt Derivation of v = v₀ + at: a = Δv / Δt Δv = at v - v₀ = at v = v₀ + at Derivation of Δx = ½(v + v₀)t: v_avg = Δx / Δt Δx = v_avg * t v_avg = (v + v₀) / 2 Δx = ½(v + v₀)t CHOOSING THE RIGHT EQUATION Guidelines: Identify known and unknown variables Determine which equation contains only one unknown If multiple equations work, choose the simplest Practice: Examples: Object dropped from height (use Δx = ½at²) Car braking to stop (use v² = v₀² + 2aΔx) Rocket accelerating for a specific time (use v = v₀ + at) PROBLEM SOLVING WITH EQUATIONS OF MOTION Step-by-Step Strategy: List given information and identify unknowns Choose appropriate equation Rearrange equation to solve for unknown Substitute known values and calculate Check units and reasonableness of answer Worked Example: Problem: A car accelerates from 10 m/s to 30 m/s over a distance of 100 m. What is its acceleration? Solution: Given: v₀ = 10 m/s, v = 30 m/s, Δx = 100 m Unknown: a Equation: v² = v₀² + 2aΔx Rearrange: a = (v² - v₀²) / (2Δx) Substitute: a = ((30 m/s)² - (10 m/s)²) / (2 * 100 m) Calculate: a = 4 m/s² Check: Units are correct, magnitude is reasonable for a car PRACTICE 2 : SOLVE PROBLEMS USING EACH EQUATION Problem 1: A car accelerates from 10 m/s to 30 m/s in 5 seconds. What is its acceleration? Problem 2: A train slows down from 25 m/s to 10 m/s over a distance of 150 m. How long does this take? Problem 3: A rocket accelerates upward at 15 m/s² from rest. How high does it go in 10 seconds? Problem 4: A cyclist brakes from 8 m/s to a complete stop. If the deceleration is 2 m/s², what distance does the cyclist cover while stopping? FREE FALL MOTION Definition: Motion under the influence of gravity alone Assumptions: Neglect air resistance Acceleration is constant (g = 9.8 m/s² downward) Key Points: Applies to both upward and downward motion For objects thrown upward: Time to reach maximum height equals time to fall back Velocity at maximum height is zero Equations (replace a with g): v = v₀ - gt (negative because g is downward) y = y₀ + v₀t - ½gt² v² = v₀² - 2g(y - y₀) Example : Object dropped from rest: After 2 seconds: v = 0 - 9.8 m/s² * 2 s = -19.6 m/s Object thrown upward at 20 m/s: Time to reach max height: 0 = 20 m/s - 9.8 m/s² * t t = 2.04 s GRAPHICAL ANALYSIS OF FREE FALL Position-Time Graph: Parabolic shape Vertex at maximum height (for objects thrown upward) Velocity-Time Graph: Straight line with negative slope y-intercept is initial velocity x-intercept (if present) indicates time at maximum height Acceleration-Time Graph: Horizontal line at -9.8 m/s² REAL-WORLD APPLICATIONS Sports: Long jump: Calculate maximum distance based on initial velocity and angle High jump: Determine minimum velocity needed to clear bar Pole vault: Analyze conversion of kinetic to potential energy Transportation: Vehicle stopping distances: Design safe following distances Aircraft takeoff: Calculate minimum runway length Train acceleration: Design comfortable acceleration for passengers Engineering: Elevator design: Ensure comfortable and safe acceleration/deceleration Amusement park rides: Design thrilling yet safe acceleration profiles Crash tests: Analyze vehicle safety based on deceleration rates Space Exploration: Rocket launches: Calculate fuel requirements based on desired acceleration Planetary landing missions: Design landing sequences considering different g values Orbital mechanics: Understand satellite motion and positioning PROBLEM-SOLVING PRACTICE Sample Problems A car accelerates from rest to 100 km/h in 8.0 s. Calculate its acceleration. A stone is thrown vertically upward with an initial velocity of 20 m/s. How high does it go? A ball is dropped from a height of 45 m. How long does it take to hit the ground? A rocket accelerates at 30 m/s² for 2.0 minutes. What is its final velocity? PREVIEW OF NEXT LESSON AND HOMEWORK Next class: Motion in two dimensions, projectile motion Chapter 1 test on Friday Sep 13th CHAPTER 2 MOTION IN TWO DIMENSIONS INTRODUCTION TO MOTION IN TWO DIMENSIONS "Understanding Vector Representations" Explanation: In our previous studies, we focused on motion in one dimension. However, real- world motion often occurs in two or even three dimensions. This lesson introduces the tools and concepts we need to describe and analyze two-dimensional motion, with a focus on vector representations. Historical context: The study of vectors dates back to the late 19th century, with contributions from mathematicians and physicists like William Rowan Hamilton and Josiah Willard Gibbs. SCALARS VS VECTORS Scalar: A quantity with only magnitude (size) Examples: Distance (5 km): The length of a path traveled, regardless of direction. Speed (60 km/h): How fast an object is moving, without specifying direction. Mass (2 kg): The amount of matter in an object, independent of gravity. Temperature (20°C): A measure of thermal energy, not associated with direction. Scalars are fully described by a single number and its unit. They are useful for many calculations but can't capture directional information.  Vector: A quantity with both magnitude and direction Examples: Displacement (5 km east): Net change in position, specifying both distance and direction. Velocity (60 km/h north): Speed in a specific direction. Acceleration (2 m/s² downward): Rate of change of velocity, including direction of change. Force (10 N upward): A push or pull with both strength and direction. Explanation: Vectors require both a number with unit and a direction to be fully described. They are essential for describing motion and forces in physics. A scalar as a simple number on a number line, and a vector as an arrow with length and direction on a coordinate plane. REAL-WORLD VECTOR EXAMPLES Displacement: Moving 5 km east This vector tells us not just how far you've moved (5 km), but also in which direction (east). If you then moved 3 km west, your total displacement would be different from your total distance traveled. Real-world application: GPS systems use displacement vectors to calculate your position relative to satellites. Velocity: Driving 60 km/h north This vector describes both the speed of the car and the direction it's traveling. It's different from speed alone because it specifies the direction of motion. Real-world application: Weather forecasts use velocity vectors to show wind direction and speed. Force: Pushing a box with 20 N of force upward This vector indicates both the strength of the push and its direction. Forces are always vector quantities because their effects depend on both magnitude and direction. Real-world application: Engineers use force vectors to analyze structures and design machines. DESCRIBING DIRECTION IN TWO DIMENSIONS Introduction to the compass rose The compass rose is a tool used to describe directions precisely. It originated in nautical charts and has been used for centuries in navigation. The term "compass rose" dates back to the 1300s when compass cards were decorated with elaborate rose designs. Primary directions: North, South, East, West These are the four main cardinal directions, corresponding to the four main points of a compass. Mnemonic device: "Never Eat Shredded Wheat" can help remember the order clockwise around the compass. Secondary directions: Northeast, Southeast, Southwest, Northwest These are midway between the primary directions, also known as ordinal or intercardinal directions.  How to express directions: [E 20° N] means "point east, then turn 20° toward north" This notation allows us to describe any direction precisely. It's particularly useful in navigation and physics problems. Mathematical perspective: This system effectively uses a polar coordinate system, with east as the 0° reference direction. Practice: Interpret [S 45° W], [N 30° E] [S 45° W] means "point south, then turn 45° toward west". This is equivalent to southwest. [N 30° E] means "point north, then turn 30° toward east". How this system relates to the degrees on a circle (0° to 360°) and how it's used in fields like surveying and orienteering.  Complementary angles: [E 20° N] is the same as [N 70° E] These two notations describe the same direction. The angles add up to 90° because they're complementary. This is useful for converting between different ways of expressing the same direction. Geometric interpretation: Draw a right triangle to illustrate how these two angles relate.Convention: North and East are positive, South and West are negative This convention helps when we start doing calculations with vectors. It aligns with the standard Cartesian coordinate system where the positive y-axis points north and the positive x-axis points east. Application: In vector component calculations, northward and eastward components will be positive, while southward and westward components will be negative. Examples of equivalent notations: [W 40° S] = [S 50° W] [E 60° S] = [S 30° E] Explanation: Practice converting between these equivalent forms to become comfortable with the notation. Draw each example on a compass rose to visualize the equivalence. Exercise: Have students try to find other equivalent pairs. VECTOR SCALE DIAGRAMS Definition: A visual representation of vectors drawn to a specific scale Purpose: To accurately represent and analyze vector quantities graphically Explanation: Scale diagrams allow us to visualize vectors that might be too large or small to draw at actual size. They're especially useful for adding vectors graphically. Components: 1. Arrow: Shows direction : The arrow points in the direction of the vector. The tip of the arrow is the head, and the other end is the tail. 2. Length: Represents magnitude: The length of the arrow is proportional to the magnitude of the vector. 3. Scale: Relates diagram to real world : The scale tells us how the length on paper relates to the actual magnitude. For example, 1 cm on paper might represent 10 m in reality. CHOOSING AN APPROPRIATE SCALE Consider the size of your paper and the vector magnitudes : Your scale should allow the vector to fit on the page while being large enough to measure accurately. If dealing with multiple vectors, ensure all can fit on the page with the same scale. Example: If your longest vector is 500 m and your paper is 30 cm wide, a scale of 1 cm : 20 m would work well. Aim for diagrams that are about half to one full page in size : This size range typically provides a good balance between detail and manageability. Larger diagrams allow for more precise measurements but may be unwieldy. Example scales: 1 cm : 10 m, 1 cm : 50 km, 1 cm : 5 N : The left side is always 1 cm on paper, the right side is the real-world equivalent. Choose a scale that makes your numbers easy to work with. Tip: Powers of 10 often make for convenient scales (e.g., 1 cm : 100 m rather than 1 cm : 90 m). Practice: Choose appropriate scales for 500 m, 2000 km, 50 N Possible answers: 1 cm : 50 m, 1 cm : 200 km, 1 cm : 10 N : These scales would result in diagrams of reasonable size for each quantity. Discuss why each scale is appropriate and what the resulting diagram sizes would be. Additional considerations: Discuss how to handle very large or very small quantities, and when it might be necessary to use different scales for different components of a problem. DRAWING VECTOR SCALE DIAGRAMS - STEP 1 Choose the scale Explanation: Based on the principles discussed in the previous slide, select a scale that will work for all vectors in your problem. Convert the real-world magnitude to diagram length : This step translates the actual vector magnitude into a length you can draw on paper. Example: Scale 1 cm : 10 m Vector: 41 m [E 20° N] Calculation: 41 m ÷ (10 m/cm) = 4.1 cm : We divide the real-world magnitude by the scale factor to get the diagram length. Tip: Always carry an extra digit in your calculations to minimize rounding errors. DRAWING VECTOR SCALE DIAGRAMS - STEP 2 Draw a Cartesian coordinate system : This provides a reference frame for your vector. The origin (0,0) is typically placed at the bottom left of your paper. The positive x-axis points east, and the positive y-axis points north. Tip: Use light pencil lines for the axes so they don't overshadow your vector. Use a ruler to draw the vector with the correct length : Precise measurement is crucial for accurate representation. Place the ruler's 0 mark at the origin and extend it in the general direction of the vector. Best practice: Hold the ruler firmly and use a sharp pencil for the most accurate line. Use a protractor to get the correct angle : Align the protractor's 0° line with east (positive x- axis), then measure 20° towards north. Technique: Place the center of the protractor at the origin of your vector (usually the tail). Ensure the 0° line of the protractor aligns perfectly with the positive x-axis. STEP-BY-STEP VISUAL GUIDE FOR THE 41 M [E 20° N] EXAMPLEDRAW THE AXES 1. Measure 4.1 cm from the origin 2. Set the protractor to 20° from east towards north 3. Draw the vector Explanation: Show each step visually, emphasizing proper tool use and measurement techniques. Common errors to avoid: Misaligning the protractor Not starting the vector at the origin Forgetting to convert the real-world magnitude to diagram length DRAWING VECTOR SCALE DIAGRAMS - STEP 3 Label the vector (magnitude, direction, scale) Explanation: Clear labeling is essential for understanding and using the diagram. Include: The vector's magnitude and units (e.g., 41 m) The direction ([E 20° N]) The scale used (1 cm : 10 m) Tip: Place labels neatly near the vector without cluttering the diagram. Add arrowhead to indicate direction Explanation: The arrowhead should be at the end point of the vector, pointing in the direction of the vector. Make it clear but not overly large. Best practice: Draw the arrowhead last to ensure it doesn't interfere with measurements. COMPLETE EXAMPLE OF 41 M [E 20° N] VECTOR DIAGRAM Show the finished diagram with all components labeled. This serves as a model for students to follow in their own work. Quality check: Encourage students to ask yourself: Is my diagram neat and clear? Have I included all necessary labels? Is the scale clearly indicated? Does the vector accurately represent the given information? PRACTICE DRAWING VECTORS Problem 1: 75 km [N 30° W], scale 1 cm : 25 km Solution walk-through: Convert 75 km to diagram length: 75 ÷ 25 = 3 cm Draw axes, measure 3 cm from origin Use protractor to measure 30° west of north Draw vector and add labels Problem 2: 120 m [S 60° E], scale 1 cm : 20 m Solution walk-through: Convert 120 m to diagram length: 120 ÷ 20 = 6 cm Draw axes, measure 6 cm from origin Use protractor to measure 60° east of south Draw vector and add labels Problem 3: 500 N [E 45° N], scale 1 cm : 100 N Solution walk-through:Convert 500 N to diagram length: 500 ÷ 100 = 5 cm 1. Draw axes, measure 5 cm from origin 2. Use protractor to measure 45° north of east 3. Draw vector and add labels For each problem, provide a step-by-step demonstration. MEASURING VECTOR SCALE DIAGRAMS Use ruler to measure length, convert back to real-world values : Measure from the tail to the tip of the vector arrow. Multiply the measured length by the scale factor to get the real-world magnitude. Example: If you measure 3.6 cm on a diagram with scale 1 cm : 15 m, the real- world magnitude is 3.6 × 15 = 54 m Use protractor to measure angle from reference direction : Place the protractor's center point at the tail of the vector. Align the 0° line with the positive x-axis (east), then read the angle to the vector. Tip: Always measure the smallest angle to the x-axis, then determine the quadrant to express the direction correctly. Example: Measure a vector 3.6 cm long at 35° from east Scale 1 cm : 15 m Real-world: 3.6 × 15 = 54 m [E 35° N] Walk through the measurement and calculation process step-by-step. ADDING VECTORS USING SCALE DIAGRAMS This lesson introduces vector addition through scale diagrams. The use of diagrams helps visualize how vectors combine, leading to an understanding of how physical quantities like force and displacement interact. We’ll explore how to combine vectors visually, a foundational method for solving problems in physics involving motion, forces, and more. REVIEW OF VECTOR BASICS A vector is a quantity that has both magnitude and direction, such as displacement, velocity, and force. Vectors are represented by arrows, with the length corresponding to magnitude and the direction showing the angle. Scalars vs. Vectors: Scalars have magnitude only (e.g., temperature), while vectors include direction (e.g., velocity). Identify which of the following are vectors: speed, force, temperature, velocity, mass. Importance:Vectors are critical for describing real-world phenomena like motion and forces, where direction matters as much as magnitude. WHY ADD VECTORS? Real-World Examples: River Crossing Problem: Imagine a boat traveling across a river with a current. The boat's velocity and the river's current are vectors that must be added to determine the actual path. Forces on an Object: Multiple forces acting on an object combine to form a resultant force, which determines the object’s motion. Wind Effects on Flights: A plane’s airspeed and wind velocity combine to determine the plane’s actual ground speed and direction. How does vector addition apply to everyday activities such as walking diagonally across a field or steering a car around a corner? THE CONCEPT OF VECTOR ADDITION Definition of Vector Addition: Vector addition involves combining two or more vectors to determine their cumulative effect, known as the resultant vector. This is essential for analyzing motions in two dimensions. Real-world example: Imagine a boat crossing a river with a strong current. The boat's velocity relative to the water combines with the velocity of the river current to give the boat's actual velocity relative to the riverbank. This is vector addition: the boat's velocity and the current’s velocity are added to determine the resultant velocity. THE TIP-TO-TAIL METHOD In the tip-to-tail method of vector addition: Draw the first vector with the correct magnitude and direction. Start the second vector at the tip (end) of the first vector. The resultant vector is drawn from the tail of the first vector to the tip of the second vector. Example: If vector A represents a car driving 4 m east and vector B represents a person walking 3 m north, you can add these using the tip-to-tail method. The resultant vector represents the total displacement. Commutative Property: Vector addition is commutative, meaning the order does not matter. A + B = B + A. STEP-BY-STEP PROCESS FOR ADDING TWO VECTORS 1. Choose an appropriate scale: If dealing with large vectors (e.g., 5 km), choose a scale that fits on your paper (e.g., 1 cm = 1 km). 2. Draw the first vector: Use a ruler and protractor. Ensure accuracy in both magnitude and direction (e.g., 4 cm for 4 km and use the appropriate angle). 3. From the tip of the first vector, draw the second vector: Position the second vector’s tail at the tip of the first vector. 4. Draw the resultant vector: The resultant vector is drawn from the tail of the first vector to the tip of the second vector. It represents the combined effect of both vectors. 5. Measure the resultant vector: Use a ruler to measure its length and a protractor for the direction (angle). 6. Convert back to real-world values: Multiply the measured length by your scale to get the magnitude in real-world units. Example: Vector A = 3 cm (representing 3 m east), and Vector B = 4 cm (representing 4 m north). After drawing these tip-to-tail, measure the resultant vector (should be approximately 5 cm using the Pythagorean theorem). WORKED EXAMPLE 1 - ADDING PERPENDICULAR VECTORS Given: Vector A = 3 m [E],Vector B = 4 m [N] Step-by-Step Process: Choose a scale: Let’s use 1 cm = 1 m. Draw Vector A: Draw a 3 cm line pointing east (right). Draw Vector B from the tip of A: Draw a 4 cm line pointing north (up) from the tip of Vector A. Draw the resultant vector: From the tail of A to the tip of B, draw the diagonal line. Measure the resultant: The diagonal will measure 5 cm (5 m in real-world values) using Pythagoras' theorem:R=(32+42)=5 mR=(32+42)=5 m Find the angle: Use trigonometry to find the angle between the resultant vector and east:θ=tan⁡ −1(43)≈53∘ north of eastθ=tan−1(34)≈ WORKED EXAMPLE 2 - ADDING NON- PERPENDICULAR VECTORS Given: Vector C = 5 km [N 30° E], Vector D = 3 km [E 20° N] Step-by-Step Process: Choose a scale: Use 1 cm = 1 km. Draw Vector C: At a 30° angle from the north, draw a 5 cm line. Draw Vector D: From the tip of Vector C, draw a 3 cm line at a 20° angle from east. Draw the resultant vector: The resultant connects the tail of C to the tip of D. Measure the resultant: The length represents the magnitude of the resultant vector. Find the angle: Use trigonometry or a protractor to determine the direction of the resultant vector. In this case, since the vectors are not perpendicular, you will need to apply the law of cosines or use a graphical method to measure the resultant vector and its angle. PRACTICE PROBLEM Problem: Add the following vectors: Vector X = 6 m [S] Vector Y = 4 m [W] Step-by-Step Process: Choose a scale (1 cm = 1 m). Draw Vector X 6 cm pointing south. From the tip of X, draw Vector Y 4 cm pointing west. Draw the resultant vector from the tail of X to the tip of Y. Measure the length and use trigonometry to find the angle. Expected answer: The resultant vector has a magnitude of approximately 7.2 m and a direction of around 33.7° west of south. ADVANTAGES AND LIMITATIONS OF THE SCALE DIAGRAM METHOD Advantages: Visualization: Provides a clear, visual way to understand the relationship between vectors. Ease of Use: Simple to perform for two or three vectors. Applicable to Various Situations: Useful for analyzing displacement, forces, and velocities in many physical scenarios. Limitations: Limited Accuracy: The precision of the resultant vector depends on the drawing's accuracy, which can be affected by the scale and manual measurement errors. Complexity with More Vectors: Adding multiple non-perpendicular vectors can become cumbersome. Not Ideal for Very Large Vectors: As scales get smaller to fit on paper, accuracy decreases. SUMMARY AND HOMEWORK ASSIGNMENT Recap key points: We’ve learned how to use scale diagrams and the tip-to-tail method to add vectors. Worked through examples involving perpendicular and non-perpendicular vectors. Highlighted the advantages and limitations of scale diagrams. Homework: Complete problems from Section 2.1 in your textbook. Focus on adding vectors using both perpendicular and non-perpendicular scenarios(Nelson physics textbook…). Additional question: Combine vectors A (10 m [N 60° E]) and B (8 m [E 30° N]) using scale diagrams. TWO-DIMENSIONAL MOTION: ALGEBRAIC APPROACH AND PROJECTILE MOTION Overview: Today, we’ll explore the mathematical analysis of two-dimensional motion. We'll break down vectors into components, add them algebraically, and apply these concepts to analyze the motion of projectiles, like sports balls and rockets. REVIEW OF VECTOR ADDITION (GRAPHICAL METHOD) Recap: Tip-to-Tail Method Vectors are quantities with both magnitude and direction. Tip-to-Tail Rule: To add two vectors, place the tip of the first vector at the tail of the second. The resultant vector goes from the tail of the first vector to the tip of the last vector. Example: Add 3m [E] and 4m [N]. Draw the 3m vector eastward, then draw the 4m vector northward from the tip of the first. The resultant vector points diagonally. Limitations of Graphical Methods: Imprecise: For large or small vectors, accurately drawing to scale becomes difficult. Complexity: Adding multiple vectors graphically is time- consuming and prone to error. Limited Use for Calculations: Hard to derive precise mathematical results like angles or magnitudes. INTRODUCTION TO VECTOR COMPONENTS Definition: A vector can be decomposed into two parts: the x- component (horizontal) and the y- component (vertical). These components describe the movement along the x- axis and y-axis, respectively. Real-World : Imagine walking in a city grid. To reach your destination, you need to walk east and then north. Your overall movement (diagonal) can be broken into an eastward (x) and northward (y) part. BREAKING DOWN VECTORS INTO COMPONENTS Using Trigonometric Functions: To find the x and y components of a vector, use sine and cosine based on the angle relative to the horizontal axis: sine (sin θ) = opposite / hypotenuse cosine (cos θ) = adjacent / hypotenuse Formulas: x-component: magnitude × cos(θ) (projects the vector onto the x-axis) y-component: magnitude × sin(θ) (projects the vector onto the y-axis) EXAMPLE: VECTOR COMPONENTS Problem: Find the components of a vector 30.0 m [E 25° N]. Solution Steps: Magnitude: 30.0 m Angle: 25° x-component = 30.0 × cos(25°) = 27.19 m [E] y-component = 30.0 × sin(25°) = 12.68 m [N] Show this vector on a coordinate plane, breaking it into the two components, one eastward and one northward. ADDING VECTORS USING COMPONENTS Process: Break each vector into its x- and y-components. Add x-components to find the total horizontal displacement. Add y-components to find the total vertical displacement. Find the resultant magnitude: √(x² + y²) Find the direction: θ = tan⁻¹(y/x) Example: Add 5m [E] and 3m [N] x-components: 5 + 0 = 5 y-components: 0 + 3 = 3 Magnitude: √(5² + 3²) = 5.83 m Direction: tan⁻¹(3/5) = 30.96° EXAMPLE: VECTOR ADDITION BY COMPONENTS Problem: Add Δd1 = 20.0 m [W] and Δd2 = 10.0 m [S 40° E]. Solution: Δd1 components: x = -20.0 m, y = 0 m Δd2 components: x = 10.0 cos(40°) = 7.66 m [E] y = -10.0 sin(40°) = -6.43 m [S] Sum of x-components: -20.0 + 7.66 = -12.34 m Sum of y-components: 0 + (-6.43) = -6.43 m Magnitude: √((-12.34)² + (-6.43)²) = 13.92 m Direction: tan⁻¹(-6.43 / -12.34) = 27.5° south of west Final Answer: 13.92 m [W 27.5° S] PRACTICE PROBLEM: COMPONENT METHOD Problem: Add 15 m [N 30° E] and 20 m [E 45° S]. Instructions: Break down each vector into its components. Add the components. Calculate the resultant’s magnitude and direction. ADVANTAGES OF COMPONENT METHOD Precision: No reliance on drawing accuracy, as calculations give exact results. Example: Add vectors like 156.78 m [N 23.45° W]. Handling Multiple Vectors: Simple process for adding more vectors by summing all x and y components. Example: Show addition of three or more vectors in complex cases. Suitability for Real Problems: Use in navigation, engineering, and physics problems like river crossing or wind resistance. I NT RO DU CT I ON TO PRO J E C TI LE M OT I ON Definition: Projectile motion is the motion of an object under the influence of gravity only (after launch). Key Idea: The motion has two independent parts: horizontal (constant velocity) and vertical (accelerated by gravity). Examples: Sports: Basketball shots follow a parabolic path. Real-World: Cannonballs and other projectiles. Modern Application: Trajectories in video games like Angry Birds. K E Y C O NC E PTS I N PR OJ E C TI L E M OT I ON 1. Independence of Motion: Horizontal and vertical motions act independently. 2. Constant Horizontal Velocity: Objects maintain their initial horizontal speed throughout flight. 3.Vertical Motion with Gravity: Vertical motion is accelerated due to gravity (g = 9.8 m/s²). ANALYZING PROJECTILE MOTION Equations for Motion: Horizontal motion: x = v₀ₓt vₓ = v₀ₓ (constant) Vertical motion: y = y₀ + v₀ᵧt - ½gt² vᵧ = v₀ᵧ - gt Key Formulas: Time of Flight: t = 2v₀ᵧ / g Range: R = (v₀² sin(2θ)) / g Max Height: h = (v₀ᵧ²) / (2g) EXAMPLE: HORIZONTAL PROJECTILE LAUNCH Problem: A ball is launched horizontally from a height of 10.0 m at 3.0 m/s. Solution: Time of Flight: t = √(2h / g) = √(2 × 10 / 9.8) = 1.43 s Range: x = v₀ₓ × t = 3.0 × 1.43 = 4.29 m PROJECTILE LAUNCHED AT AN ANGLE Components of Initial Velocity: v₀ₓ = v₀ cos(θ) v₀ᵧ = v₀ sin(θ) Trajectory Shape: The path is a parabola due to the combination of constant horizontal velocity and accelerating vertical motion. EXAMPLE: ANGLED PROJECTILE LAUNCH Problem: A ball is launched at 45° with an initial speed of 20 m/s. Solution: Maximum Height: h = (v₀ sin(θ))² / (2g) = 10.2 m Range: R = (v₀² sin(2θ)) / g = 40.8 m APPLICATIONS OF PROJECTILE MOTION Sports: Shot Put: Optimizing throw angle and speed. Long Jump: Maximizing range through launch angle. Engineering: Fountains: Designing specific spray patterns. Firefighting: Calculating water jet trajectories. Ballistics: Firearm forensics. Missile trajectory prediction. COMMON MISCONCEPTIONS IN PROJECTILE MOTION 1. Effect of Mass on Trajectory: Misconception: Heavier objects fall faster. Reality: All objects fall at the same rate in the absence of air resistance. 2. Angle vs. Range: Misconception: 45° always gives maximum range. Reality: This is true only on flat ground without air resistance. SUMMARY Vector Component Method: Break vectors into x and y components. Add the components separately, then reconstruct the resultant. Projectile Motion: Horizontal and vertical motions are independent. The trajectory is parabolic, influenced by launch speed, angle, and gravity. HOMEWORK ASSIGNMENT Problems from Textbook: Section 2.2: Questions 3, 5, 7 Section 2.3: Questions 2, 4, 6 "PROJECTI LE MOTI ON AND HI STORICAL PERSP ECTI VES" "To understand the motion of objects in our universe, we must first understand motion on Earth." REVIEW OF PROJECTILE MOTION BASICS Definition: A projectile is any object that moves through the air or space, acted on only by gravity (and air resistance, if applicable) Independence of horizontal and vertical motion: Horizontal motion: Constant velocity (neglecting air resistance) Vertical motion: Constant acceleration due to gravity Examples: Baseball throw: Discuss how a pitcher's throw combines horizontal and vertical components Cannonball: Historical context of ballistics in warfare Rocket launch: Modern application, discussing stages of flight KEY EQUATIONS FOR PROJECTILE MOTION Horizontal motion: x = v₀ₓt, vₓ = v₀ₓ Explanation: No acceleration in horizontal direction, velocity remains constant Vertical motion: y = y₀ + v₀ᵧt - ½gt², vᵧ = v₀ᵧ - gt Explanation: Constant acceleration due to gravity, velocity changes linearly with time Importance: These equations allow us to predict the position and velocity of a projectile at any time Example: Dropping a ball from a height ANALYZING Example problem: A ball is launched horizontally from a cliff 45 m high with an initial velocity of 15 m/s. PROJECTILE Step-by-step solution: LAUNCHED Time of flight: t = √(2h/g) = √(2*45/9.8) = 3.03 s Explanation: This comes from vertical motion equation, setting final y position to 0 HORIZONTALLY Horizontal distance: x = v₀ₓt = 15 * 3.03 = 45.45 m Explanation: Constant horizontal velocity throughout flight Real-world application: Discuss how this applies to long jump in athletics PROJECTILES LAUNCHED AT AN ANGLE Components of initial velocity: v₀ₓ = v₀ cos(θ), v₀ᵧ = v₀ sin(θ) Explanation: Use trigonometry to break velocity into components Maximum height: h_max = (v₀ᵧ)²/(2g) Derivation: Show how this comes from vertical motion equations Time of flight: t_total = 2v₀ᵧ/g Explanation: Time to reach max height, doubled for total flight time Parabolic nature of the trajectory explained Visual: Interactive graph showing how changing angle affects trajectory shape Example: Ski jump Discuss how skiers optimize their jump angle and initial velocity RANGE OF A PROJECTILE Equation for range: R = (v₀² sin(2θ))/g Angle for maximum range: 45° (in ideal conditions) Show mathematically why 45° gives maximum range Discussion on how air resistance affects this in real-world scenarios Example: Compare theoretical and actual ranges for a golf drive EXAMPLE PROBLEM: ANGLED PROJECTILE Problem: A soccer ball is kicked with an initial velocity of 20 m/s at an angle of 30° above the horizontal. Solution: Max height: h_max = (v₀ sin(θ))²/(2g) = 3.19 m Time of flight: t_total = 2(v₀ sin(θ))/g = 2.04 s Range: R = (v₀² sin(2θ))/g = 35.4 m FACTORS AFFECTING PROJECTILE MOTION Initial velocity: Effect on range and height Example: Compare trajectories of a baseball pitched at different speeds Launch angle: Optimal angles for different purposes Example: Discuss differences in optimal angle for basketball shot vs. javelin throw Air resistance: Introduction to real-world complications Visual: Graph comparing ideal trajectory to one with air resistance Gravity:Variation on different planets Example: Calculate how far you could throw a ball on the Moon vs. Earth APPLICATIONS OF PROJECTILE MOTION Sports: Basketball: Optimizing free throw trajectory Football: Calculating best angle for a long pass Golf: Analyzing drive distance and factors affecting it Engineering: Fountain design: Creating specific water patterns Catapults: Historical and modern uses (e.g., aircraft carrier launchers) Rocket launches: Multi-stage trajectory planning Military: Ballistics calculations for artillery Entertainment: Physics engines in video games (e.g., Angry Birds, Worms)  I NT RO DU CT I ON TO H I S TOR I CA L PE RS PE C TI V E S Importance of understanding the development of scientific ideas How our understanding of motion evolved over centuries GALILEO GALILEI: SIXTEENTH-CENTURY "NEW SCIENTIST" Life: Born in 1564 in Pisa, Italy Visual: Map of Renaissance Italy, highlighting Pisa and Florence Education and early career: Studies in medicine, shift to mathematics and physics Major works: "Two New Sciences" - Discuss its revolutionary ideas Conflict with the Church: Heliocentrism controversy Example: Compare Ptolemaic and Copernican models of the solar system https://www.youtube.com/watch?v=JI36dazqEwU GALILEO'S EXPERIMENTS Leaning Tower of Pisa experiment: Legend vs. historical evidence Importance of challenging established beliefs Visual: Artistic depiction of the legendary experiment https://www.youtube.com/watch?v=LPG8XNFLRsk Inclined plane experiments: How it allowed for precise measurements of motion Replication: Discuss modern recreations of Galileo's experiments https://www.youtube.com/watch?v=Axv5m7E0hkc&t=5s GALILEO'S DISCOVERIES Objects fall at the same rate regardless of mass: Contradiction to Aristotelian physics Importance of air resistance in real-world scenarios Example: Calculate fall times for a feather and a hammer in vacuum vs. air Concept of inertia: Objects in motion tend to stay in motion Precursor to Newton's First Law Example: Discuss how this concept applies to seat belts in cars https://www.youtube.com/watch?v=wL9XopHoevU IMPACT OF GALILEO'S WORK Challenge to Aristotelian physics: Shift from philosophy-based to experiment-based science Example: Compare Aristotle's and Galileo's approaches to understanding motion Foundation for Newton's laws: How Galileo's work on motion and inertia influenced Newton Visual: Connect Galileo's ideas to Newton's laws with a concept map Improvements in scientific instruments and methods Example: Discuss Galileo's improvements to the telescope THE SCIENTIFIC METHOD Galileo's role in developing the experimental approach: Emphasis on measurement and mathematical description Importance of controlled experiments Example: Contrast Galileo's approach with earlier natural philosophers Modern scientific method: Observation, hypothesis, experiment, analysis, conclusion Activity: Walk through a simple experiment using scientific method steps  SUMMARY Key points from projectile motion: Independence of horizontal and vertical motion Parabolic trajectories Factors affecting projectile paths Visual: Infographic summarizing key concepts Historical significance of motion studies: Shift towards experimental science Foundation for classical mechanics Discussion: How these discoveries changed our view of the universe HOMEWORK ASSIGNMENT Problems from sections 2.3 and 2.4 in the textbook Calculate the range of a projectile launched at 60° with initial velocity 30 m/s Determine the initial velocity needed for a basketball to swish through a hoop 3 m away and 3 m high Research project: Choose a historical figure in physics (e.g., Galileo, Newton, Kepler) and write a short essay on their contributions to our understanding of motion Guidelines: 500 words, at least 3 reliable sources, discuss one experiment in detail (Next Monday) INTRODUCTION TO FORCES AND MOTION Forces are all around us, affecting every object we see Examples: Car motion: acceleration, braking, turning Bridge design: supporting weight, withstanding wind Sports: hitting a baseball, kicking a soccer ball Daily activities: opening doors, lifting objects, walking Understanding forces is essential for scientific description of our environment Today: explore different types of forces and Newton's First Law of Motion WHAT IS A FORCE? Definition: Force is a push or pull that can change the motion of an object Force is a vector quantity (has magnitude and direction) Measured in newtons (N) 1 N = 1 kg·m/s² (force required to accelerate 1 kg mass at 1 m/s²) Can be measured using: Spring scale: stretches proportionally to applied force Force sensor: electronic device providing digital reading COMMON FORCES (PART 1) Applied force (F̅ₐ): Result of direct contact between objects Example: pushing a shopping cart, kicking a ball Tension (F̅T): Pulling force exerted by a rope or string Always directed toward the rope or string Example: force in a cable supporting an elevator COMMON FORCES (PART 2) Normal force (F̅N): Perpendicular force exerted by a surface Always points away from the surface Example: force of a table supporting a book Friction (F̅f): Resists motion between surfaces in contact Acts parallel to the surface, opposite to motion or attempted motion Example: force slowing down a sliding block COMMON FORCES (PART 3) Force of gravity (F̅g): Attraction between objects due to their mass On Earth's surface, always points toward Earth's center Calculated using F̅g = mg̅, where g̅ = 9.8 m/s² [down] Example: weight of an object FORCE DIAGRAMS (PART 1) System diagram: Simple sketch of all objects involved in a situation Shows how objects interact Free-body diagram (FBD): Shows all forces acting on a single object Object represented as a dot or rectangle Forces drawn as arrows from the object FO RC E DIAG RAMS EXAMPLES: A ) BOOK ON A TABLE: System diagram: a) Show book and table FBD of book: Normal force up, gravity down b) Person pulling a sled: System diagram: Person, rope, sled, ground FBD of sled: Tension, normal force, friction, gravity c) Ball falling through air: System diagram: Ball and air FBD of ball: Gravity down, air resistance up (if considered)  Net force (F̅net): The vector sum of all forces acting on an object CALCULATING NET FORCE Steps to calculate: (PART 1) Draw FBD Choose positive direction Add forces, considering direction Remember: Forces in same direction add, opposite directions subtract CALCULATING NET FORCE (PART 2) Example problems: a) Box pushed by two people in same direction: Person 1 pushes with 50 N [E], Person 2 with 30 N [E] F̅net = 50 N + 30 N = 80 N [E] b) Tug-of-war scenario: Team A pulls with 500 N [E], Team B pulls with 450 N [W] F̅net = 500 N + (-450 N) = 50 N [E] FUNDAMENTAL FORCES (PART 1) Gravitational force: Attraction between all objects with mass Responsible for weight, planetary orbits Weakest of fundamental forces, but infinite range  FUNDAMENTAL FORCES (PART 2) Electromagnetic force: Interaction between charged particles Responsible for chemical bonds, electric currents Much stronger than gravity, infinite range Strong nuclear force: Holds protons and neutrons together in nucleus Strongest force, but very short range Weak nuclear force: Responsible for radioactive decay Weaker than strong force, very short range I NE RTI A ( PAR T 1 ) The resistance of an object to changes in its motion Inertia is directly proportional to the mass of an object Newton's first law is also called the law of inertia More mass means more inertia INERTIA (PART 2) EXAMPLES: a) Difficulty in starting to push a heavy object: More mass means more resistance to change in motion b) Tendency to keep moving forward when a car stops suddenly: Passengers continue moving due to inertia c) A tablecloth pulled quickly from under dishes: Dishes have inertia and tend to stay in place d) A hammer driving a nail: Hammer's inertia allows it to deliver force to the nail NEWTON'S FIRST LAW OF MOTION An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction, unless acted upon by an unbalanced force. Implications: a) Objects at rest tend to stay at rest b) Objects in motion tend to stay in motion c) Force is required to change velocity (speed or direction) d) If net force is zero, velocity remains constant (including zero velocity) NEWTON'S FIRST LAW OF MOTION Key points: Applies to both objects at rest and in motion Explains why friction eventually stops moving objects Helps understand the concept of equilibrium Crucial for understanding safety features in vehicles APPLICATIONS OF NEWTON'S FIRST LAW Examples in everyday life: a) Objects sliding on a car dashboard during acceleration: Objects continue moving straight while car turns b) A person continuing to move forward when a bus stops suddenly: Person's body tends to continue moving c) The need for seatbelts in vehicles: Seatbelts provide force to change motion during collisions APPLICATIONS OF NEWTON'S FIRST LAW (PART 2) d) A coin on a card over a glass (flicking the card): Coin's inertia keeps it in place while card moves e) A satellite continuing to orbit Earth: No air resistance in space, so motion continues f) Ice skater gliding on ice: Continues moving due to little friction g) Difficulty in starting to move a heavy object: Inertia of rest must be overcome PRACTICE PROBLEMS (PART 1) a) Draw FBD for a book sliding down an inclined plane b) Calculate net force on a 5 kg object with 20 N [E] and 15 N [W] forces c) Explain why a heavy box is harder to start moving than a light one d) Describe how Newton's First Law applies to a skater gliding on ice PRACTICE PROBLEMS (PART 2) e) A 2 kg book is at rest on a table. Draw the FBD and calculate the normal force. f) Two people push a 50 kg crate. One pushes with 200 N [E], the other with 150 N [W]. Calculate the net force and acceleration of the crate. g) Explain how Newton's First Law relates to the design of airbags in cars. SUMMARY AND NEXT STEPS Fand Newton's First Law Preview upcoming topics: Newton's Second and Third Laws FREE-BODY DIAGRAM PROBLEM: A 5 kg book is resting on a table. Draw a free-body diagram for the book, labeling all forces acting on it. Solution: Draw a rectangle representing the book Draw an arrow pointing downward labeled F̅g (force of gravity) Draw an arrow pointing upward labeled F̅N (normal force) The magnitude of F̅g = mg = (5 kg)(9.8 m/s²) = 49 N Since the book is at rest, F̅N must equal F̅g in magnitude, so F̅N = 49 N NET FORCE CALCULATION: A 2 kg box is being pulled to the right with a force of 10 N. A frictional force of 4 N acts to the left. Calculate the net force on the box. Solution: Define right as positive direction F̅net = F̅applied + F̅friction F̅net = 10 N + (-4 N) = 6 N [right] INTRODUCTION TO NEWTON'S SECOND AND THIRD LAWS Recap: Newton's First Law deals with objects at rest or in uniform motion Today: How forces cause changes in motion (Second Law) and interact between objects (Third Law) These laws form the foundation of classical mechanics https://www.youtube.com/watch?v=8o3j1wpabes NEWTON'S SECOND LAW - THE BASICS Statement: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass Mathematical representation: F̅net = ma̅ This law explains why a small force can move a light object easily but not a heavy one VARIABLES IN NEWTON'S SECOND LAW Force (F): measured in newtons (N) 1 N is the force required to accelerate 1 kg at 1 m/s² Mass (m): measured in kilograms (kg) Represents the amount of matter in an object Acceleration (a): measured in meters per second squared (m/s²) Rate of change of velocity ANALYZING NEWTON'S SECOND LAW If mass is constant: Increasing net force increases acceleration Example: Pushing a shopping cart harder makes it accelerate faster Decreasing net force decreases acceleration Example: Easing off the gas pedal in a car reduces acceleration If net force is constant: Increasing mass decreases acceleration Example: Adding groceries to a shopping cart makes it harder to accelerate Decreasing mass increases acceleration Example: An empty truck accelerates faster than a loaded one GRAPHICAL REPRESENTATIONS Graph of net force vs. acceleration (linear relationship) Slope represents the mass of the object Graph of acceleration vs. mass (inverse relationship) Hyperbolic curve These graphs help visualize the relationships in F̅net = ma̅ PROBLEM-SOLVING WITH NEWTON'S SECOND LAW Step-by-step approach: 1. Identify known quantities and the unknown 2. Write down F̅net = ma̅ 3. Substitute known values 4. Solve for the unknown Example problem: A 1500 kg car experiences a net force of 3000 N. Calculate its acceleration.  FREE FALL AND NEWTON'S SECOND LAW In free fall, net force = force of gravity F̅g = mg̅ where g̅ ≈ 9.8 m/s² [down] on Earth's surface All objects fall with the same acceleration in the absence of air resistance Example: A 0.5 kg ball and a 5 kg ball are dropped simultaneously. Calculate their force and compare. INTRODUCTION TO NEWTON'S THIRD LAW For every action force, there is a simultaneous reaction force that is equal in magnitude but opposite in direction Action and reaction forces act on different objects This law explains many everyday phenomena and is crucial in engineering ACTION-REACTION FORCE PAIRS EXAMPLES: A book on a table: Action: Book exerts downward force on table Reaction: Table exerts upward force on book A person walking: Action: Person pushes backward on ground Reaction: Ground pushes forward on person A hammer hitting a nail: Action: Hammer exerts force on nail Reaction: Nail exerts equal force on hammer NEWTON'S THIRD LAW IN EVERYDAY LIFE Swimming: Action: Swimmer pushes water backward Reaction: Water pushes swimmer forward Rocket propulsion: Action: Rocket expels gases downward Reaction: Gases push rocket upward Recoil in firearms: Action: Gun exerts force on bullet Reaction: Bullet exerts force on gun, causing recoil COMMON MISCONCEPTIONS ABOUT NEWTON'S THIRD LAW Misconception: Action and reaction forces cancel each other out Reality: They act on different objects, so they don't cancel Misconception: Stronger objects exert larger forces Reality: Forces are always equal, regardless of object size or strength Example: A horse pulling a cart Action: Horse pulls on cart Reaction: Cart pulls back on horse with equal force The cart moves because the horse overcomes friction, not because it exerts a larger force APPLICATIONS OF NEWTON'S THIRD LAW Rocket propulsion: Detailed explanation of how rockets work in space Conservation of momentum principle How birds fly: Wings push air down, air pushes wings up How fish swim: Fins push water backward, water pushes fish forward PRACTICE PROBLEM 1 A 1000 kg car accelerates from 0 to 100 km/h (27.8 m/s) in 10 seconds. Calculate the net force acting on the car. Solution: a = Δv/Δt = 27.8 m/s / 10 s = 2.78 m/s² F̅net = ma̅ = 1000 kg × 2.78 m/s² = 2780 N PRACTICE PROBLEM 2 A 0.145 kg baseball is hit with a bat, experiencing a force of 800 N. What is its acceleration? Solution: PRACTICE PROBLEM 3 Explain how Newton's Third Law applies when you jump off a boat onto a dock. Action: You push backward on the boat Reaction: Boat pushes forward on you This causes the boat to move away from the dock as you jump SUMMARY Newton's Second Law: F̅net = ma̅ Explains how forces cause acceleration Crucial for predicting motion in physics and engineering Newton's Third Law: Action-Reaction pairs Explains force interactions between objects Essential for understanding complex systems and designing machines REVIEW OF NEWTON'S LAWS First Law (Law of Inertia): An object remains at rest or in uniform motion unless acted upon by a net external force. Example: A book stays stationary on a table unless an external force like a push moves it. Real-world Application: Seat belts in cars act as an external force to prevent passengers from continuing in motion during a sudden stop. Second Law (F = ma): Acceleration of an object is directly proportional to the net force and inversely proportional to its mass. Example: A force of 10 N applied to a 2 kg object causes an acceleration of 5 m/s². Exercise: Calculate the acceleration when a 3 kg object is pushed with 15 N of force. Third Law (Action-Reaction): For every action force, there is an equal and opposite reaction force. Example: A swimmer pushing off the pool wall feels the wall push back, allowing them to propel forward. Exercise: Describe the action-reaction forces involved in walking. FREE BODY DIAGRAMS (FBDS) A Free Body Diagram is essential for visualizing and solving force problems. It shows the magnitude and direction of all forces acting on an object. How to draw FBDs: Represent the object as a point or box. Draw arrows for each force acting on the object. Label the forces Example: A block sliding down an incline will experience gravitational force (divided into components), normal force, and friction force. SOLVING PROBLEMS USING NEWTON’S LAWS Detailed Steps: Identify Forces: Start by listing all forces acting on the object (gravity, normal, applied force, etc.). Draw FBD: Visualize the situation with all relevant forces. Set up Coordinate System: Define positive and negative directions. Apply Newton’s Second Law: Write out the force equations (F = ma). Solve the Equations: Solve for unknown variables like force, acceleration, or tension. Example Problem: A 15 kg box is pushed with a 40 N force on a frictionless surface. What is the acceleration? (Answer: 2.67 m/s²). EXAMPLE - OBJECT ON A HORIZONTAL SURFACE Problem: A 6 kg box is pushed with a 30 N force. The coefficient of kinetic friction is 0.25. Find the acceleration. Solution: Calculate normal force: N=mg=6 kg×9.8 m/s2=58.8 NN=mg=6kg×9.8m/s2=58.8N Friction force: fk=μk×N=0.25×58.8=14.7 Nfk=μk×N=0.25×58.8=14.7N Net force: Fnet=30 N−14.7 N=15.3 NFnet=30N−14.7N=15.3N Acceleration: a=Fnet/m=15.3 N/6 kg=2.55 m/s2a=Fnet/m=15.3N/6kg=2.55m/s2 Exercise: A 10 kg box is pushed with a 50 N force on a surface with μk=0.2μk=0.2. Find the acceleration. FORCES O N AN INCLINED PLANE Explanation: On an inclined plane, forces need to be broken down into components parallel and perpendicular to the plane. Example Problem: A 4 kg block slides down a 20° incline. Find the acceleration (ignore friction). Parallel force: F∥=mgsin⁡(20∘)F∥​=mgsin(20∘) Normal force: N=mgcos⁡(20∘)N=mgcos(20∘) Acceleration: a=F∥/ma=F∥​/m Exercise: Calculate the acceleration of a 10 kg box sliding down a 15° incline with friction μk=0.3μk=0.3.  TENSIO N IN ROPES AND CABLES Expanded Explanation: Tension is the force transmitted through a string, rope, or cable when it is pulled by forces acting from opposite ends. Example: A 6 kg mass is hanging from a rope. Find the tension in the rope. T=mg=6 kg×9.8 m/s2=58.8 N CONNECTED OBJECTS System of Objects: When multiple objects are connected (by ropes, pulleys, etc.), treat them as a single system to find acceleration and tension. Example Problem: Two blocks (5 kg on a table and 3 kg hanging) connected by a pulley. Find the acceleration and tension. FBD for each mass. Write equations of motion for each block. Solve for acceleration and tension. Exercise: Repeat the example with different masses (e.g., 6 kg and 4 kg) and calculate new values. ATWOOD’S MACHINE An Atwood machine involves two masses connected over a pulley. The heavier mass accelerates downward, while the lighter mass accelerates upward. Example Problem: Two masses, 4 kg and 2 kg, are connected over a frictionless pulley. Find acceleration and tension. Net force: (4−2)g=6a(4−2)g=6a Solve for acceleration and tension. Exercise: Solve for acceleration and tension for masses of 5 kg and 3 kg. CHAPTER 4: APPLICATIONS OF FORCES Introduction: Overview: This chapter explores the fundamental forces that affect objects in motion, with a special focus on gravitational forces near Earth and the role of friction in daily life. Key Concepts: How gravitational force impacts objects on Earth and in space. The relationship between mass, weight, and gravitational field strength. The role of friction in everyday activities, such as walking, driving, and using machinery. Practical applications in automotive technology, including braking systems, tire design, and safety features. Main Objectives: Understand how forces cause changes in motion. Solve problems involving gravitational forces and friction. Analyze the impact of forces in technology, especially in vehicles. CHAPTER 4.1 - GRAVITATIONAL FORCE NEAR EARTH Air Resistance and Free Fall Free Fall: Motion where the only force acting is gravity. Example: A ball dropped from a height is in free fall until it encounters air resistance. Air Resistance Factors: Cross-sectional area: Larger areas increase resistance (e.g., a skydiver spread-eagle experiences more drag than one falling headfirst). Speed: The faster the object, the greater the air resistance. Shape: Streamlined objects (e.g., cars, planes) reduce air resistance compared to blunt shapes. Example: A feather and a hammer fall at different rates in the presence of air but fall equally in a vacuum. Terminal Speed: The constant speed when the force of air resistance equals the force of gravity. Example: Skydivers reach terminal speed before deploying parachutes, around 53 m/s (120 mph). GRAVITATIONAL FIELD STRENGTH (G) Force per unit mass experienced in a gravitational field. Units: Newtons per kilogram (N/kg), equivalent to m/s². Near Earth: g = 9.8 N/kg, meaning objects near Earth's surface accelerate at 9.8 m/s². Variation: Altitude: g decreases slightly with altitude, as seen on the International Space Station where it's around 8.7 N/kg. Latitude: g is slightly stronger at the poles than at the equator due to Earth's shape (an oblate spheroid). Example Calculation: A person with a mass of 70 kg experiences a gravitational force (weight) of 70 kg × 9.8 N/kg = 686 N on Earth. On the Moon, where g = 1.6 N/kg, the weight would be 112 N. MASS VS. WEIGHT Mass: The amount of matter in an object, constant everywhere (measured in kg). Weight: The gravitational force acting on an object, varying by location (measured in Newtons). Formula: Weight=m×gWeight=m×g Example: A 70 kg person weighs 686 N on Earth but only 112 N on the Moon. Key Differences: Mass is scalar, weight is vector (it has direction). Mass stays constant; weight changes depending on gravity. APPARENT WEIGHT Apparent Weight: How heavy you feel, influenced by forces other than gravity (e.g., acceleration). Example: An elevator ride: When accelerating upward, the normal force increases, and you feel heavier. When accelerating downward, the normal force decreases, making you feel lighter. During free-fall (if an elevator were to break), you would feel weightless as there is no normal force acting. Example : If an elevator accelerates upwards at 2 m/s², a person’s apparent weight would be calculated using F=ma. CHAPTER 4.2 - FRICTION Static vs. Kinetic Friction Static Friction: Force that prevents motion between stationary surfaces. Maximum static friction: Fs≤μsN, where μsμs is the coefficient of static friction, and N is the normal force. Kinetic Friction: Force acting between moving surfaces. Formula: Fk=μkN, where μkμk is the coefficient of kinetic friction (typically less than μsμs). Example: Pushing a heavy box—more force is needed to get it moving (static friction) than to keep it moving (kinetic friction). C OE F F I C I E NT S OF F RI C TI O N The ratio of friction force to normal force. Typical Values: Rubber on dry concrete: μs≈0.9,μk≈0.7 Ice on ice: μs≈0.1,μk≈0.03 Teflon on Teflon: μs≈0.04,μk≈0.04 COEFFICIENTS OF FRICTION Definition: The ratio of friction force to normal force. Typical Values: Rubber on dry concrete: μs≈0.9,μk≈0.7 Ice on ice: μs≈0.1,μk≈0.03 Teflon on Teflon: μs≈0.04,μk≈0.04 Example Calculations: Calculate the force required to push a crate with a known coefficient of friction and normal force. FACTORS AFFECTING FRICTION Surface Materials: Different materials interact with varying friction (e.g., rubber has more grip than metal). Surface Roughness: Rougher surfaces have more friction. Normal Force: Increased normal force increases friction. Rubber Exception: In some cases, the contact area affects friction (e.g., race cars use wide tires for better grip). Example: Off-road tires have deeper treads to improve grip on rough terrain. APPLICATIONS OF FRICTION Beneficial Friction: Walking relies on friction between shoes and the ground. Car tires grip the road for traction. Detrimental Friction: Causes wear in mechanical parts. Energy loss in engines. Reducing Friction: Lubricants, ball bearings, and air cushions. Increasing Friction: Use of tread patterns on tires, and sticky shoe soles. CHAPTER 4.3 - SOLVING FRICTION PROBLEMS Problem-Solving Steps 1. Draw a Free Body Diagram (FBD). 2. Identify known and unknown quantities. 3. Choose a coordinate system (usually +y+y up, +x+x in the direction of motion). 4. Apply Newton’s laws (ΣF=maΣF=ma). 5. Solve the resulting equations. 1. Example: A box sliding across a floor. Solve for acceleration given frictional forces. SAMPLE PROBLEM - OBJECT ON A HORIZONTAL SURFACE Scenario: 50 kg crate pushed across the floor with μk=0.2μk=0.2. Free Body Diagram (FBD): Include weight, normal force, applied force, and friction force. Equations: ΣFy=0:N=mgΣFy=0:N=mg ΣFx=ma:Fapplied−Fk=maΣFx=ma:Fapplied−Fk=ma Solution: Calculate acceleration for a given applied force. SAMPLE PROBLEM - OBJECT ON INCLINED PLANE Scenario: A 10 kg box slides down a 30° incline with μk=0.1μk=0.1. Free Body Diagram (FBD): Include weight components, normal force, and friction. Equations: Parallel: ΣF∥=mgsin(θ)−Fk=maΣF∥​=mgsin(θ)−Fk=ma Perpendicular: ΣF⊥=0:N=mgcos(θ)ΣF⊥​=0:N=mgcos(θ) Solution: Find the acceleration of the box down the plane. 1) A 1500 kg car is traveling at 25 m/s on a horizontal road when the driver sees an obstacle and applies the brakes. The coefficient of static friction between the tires and the dry road is 0.80, while the coefficient of kinetic friction is 0.60. a) Calculate the maximum deceleration possible without the wheels skidding. b) If the driver applies just enough force to reach this maximum deceleration, how far will the car travel before stopping? c) If instead the driver applies too much force and causes the wheels to lock, how far will the car travel before stopping? 2) A 75 kg skier is going down a slope inclined at 20° to the horizontal. The coefficient of kinetic friction between the skis and the snow is 0.12. a) Draw a free-body diagram for the skier. b) Calculate the acceleration of the skier down the slope. c) If the skier starts from rest, how fast will they be moving after traveling 100m down the slope? 3) A 200 g hockey puck is struck, giving it an initial velocity of 30 m/s across the ice. The coefficient of kinetic friction between the puck and the ice is 0.05. a) Calculate the acceleration of the puck as it slides across the ice. b) How far will the puck travel before coming to rest? c) If the same force that struck the puck was applied for 0.1 seconds, calculate the magnitude of this force. d) How would the puck's motion change if it was traveling on concrete instead of ice? 4) Two crates, with masses of 40 kg and 60 kg, are connected by a light rope and pulled across a horizontal floor by a force of 250 N. The coefficient of kinetic friction between the crates and the floor is 0.30. a) Draw a free-body diagram for each crate. b) Calculate the acceleration of the system. c) Determine the tension in the rope between the crates. d) If the 60 kg crate is suddenly lifted off the ground, how would the acceleration of the 40 kg crate change? 5) A 1200 kg car is traveling at 20 m/s when it begins to ascend a hill with a 5° incline. The car's engine provides a constant force of 3000 N parallel to the incline, and the coefficient of rolling friction between the tires and the road is 0.015. a) Draw a free-body diagram for the car on the incline. b) Calculate the car's acceleration as it goes up the hill. c) How far up the hill will the car travel before it comes to a stop? d) If the car reaches the top of the hill (which is 100 m long) and continues on a level road, what will its velocity be at the top of the hill? 4.4 - FORCES APPLIED TO AUTOMOTIVE TECHNOLOGY Tire Design Tread Patterns: Channels designed to displace water, prevent hydroplaning. Different patterns for various conditions (e.g., all-season, winter). Wide Tires for Racing: Greater contact area increases friction. Trade-off: Increases rolling resistance. Hydroplaning: Occurs when water builds up under tires. Prevention: Deep treads, proper tire pressure, reduced speed. BRAKING SYSTEMS Disc Brakes: Components: Rotor, caliper, and brake pads. Work through friction between pads and rotors. https://www.youtube.com/watch?v=MAuVDB-G-HQ Anti-Lock Braking System (ABS): Prevents wheel lock by rapidly applying/releasing brake pressure. Ensures shorter stopping distances and maintains steering control. https://www.youtube.com/watch?v=98DXe3uKwfc STABILITY CONTROL Traction Control: Prevents wheel spin by reducing engine power or applying brakes to spinning wheels. Electronic Stability Control (ESC): Prevents skidding by applying brakes to individual wheels to maintain control in understeer or oversteer situations. https://www.youtube.com/watch?v=FjOEy2oFFz0  UNIT 3 CHAPTER 6 ENERGY AND SO CIETY - UNIT INTROD UCT ION Understand how work, energy, and power relate to societal needs and environmental impacts. Energy Transformations: How energy changes form and the implications. Energy Efficiency and Losses: No system is 100% efficient; thermal losses are common. Social Responsibility: Balancing technology’s benefits and environmental costs. BIG IDEAS AND EXPECTATIONS OF UNIT 3 Big Ideas: Learning Expectations: Energy transformation efficiency and its societal Analyze the social/environmental effects of impacts. energy technologies. Apply energy principles in problem-solving and real-world contexts.  WHY STUDY ENERGY AND SOCIETY? Importance of Energy Canada’s Energy Landscape: Literacy: Rising demand for sustainable energy solutions. Heavy reliance on coal, oil, and nuclear energy. Understanding energy efficiency and responsible Push for cleaner energy alternatives, like wind consumption. and solar farms. CHAPTER 5.1 - WHAT IS WORK? Work Defined: Work is done when a force moves an object in its direction. Formula: W=FΔ*d. Example: A person pushes a box with 20 N force across a 5 m floor. W=20×5=100W=20×5=100 J. Exercise: If the same 20 N force pushes the box for 10 m, how much work is done? (Answer: 200 J)  POSITIVE AND NEGATIVE WORK Positive Work: Force and displacement are in the same direction. Negative Work: Force and displacement are in opposite directions. Example: A shopper pushes a cart forward with an applied force of 41 N over 11 m, and friction exerts 35 N in the opposite direction. Total work = 41×11−35×11=66. Exercise: A person lifts a 10 kg box upward, but gravity exerts a downward force. Calculate the work done by gravity over 2 m. (Answer: -196 J)  WORK DONE AT ANGLES W=F(cosθ)Δx. Example: A vacuum is pushed at a 30° angle with 50 N force over 3 m. Calculation: W=50cos(30°)×3=130 J. Exercise: A lawnmower is pushed at a 45° angle with 100 N force over 6 m. Calculate the work done. (Answer: 424.3 J) EXAMPLE : LIFTING A BUCKET A construction worker lifts a 15 kg bucket of concrete vertically from the ground to a height of 2.5 meters. Given: Mass of the bucket, m=15 kgm=15kg Height lifted, h=2.5 mh=2.5m Gravitational acceleration, g=9.8 m/s2 The force exerted to lift the bucket is F=mg=15×9.8=147 Work done to lift the bucket: W=FΔd=147×2.5=367.5 J The construction worker does 367.5 J of work to lift the bucket. EXAMPLE : PULLING A SLED AT AN ANGLE A person pulls a sled with a 30 N force at a 25° angle to the horizontal for 15 meters. Given: Force, F=30 N Angle, θ=25∘ Displacement, Δd=15 m Calculation: Work done: W=Fcos(θ)Δd=30cos(25∘)×15≈407.7 J Conclusion: The person does approximately 407.7 J of work in pulling the sled. EXAMPLE : CARRYING A BACKPACK UP STAIRS A student carries a 10 kg backpack up a flight of stairs that is 4 meters high. Given: Mass of the backpack, m=10 kg Height of stairs, h=4 m Gravitational acceleration, g=9.8 m/s2 The force exerted is the weight of the backpack: F=mg=10×9.8=98 N Work done: W=FΔd=98×4=392 J The student does 392 J of work in carrying the backpack up the stairs. CHAPTER 5.2 - WHAT IS ENERGY? Energy Defined: The ability to do work or cause change. Key Forms: Kinetic Energy: Energy of motion. Potential Energy: Stored energy based on position. KINETIC ENERGY - CALCULATION AND EXAMPLES 1 Ek=2 × 𝑚 × 𝑣 2 Example: A 0.15 kg baseball moving at 35 m/s. Calculation: Ek=0.5×0.15× 352 =92j A 70 kg cyclist moves at 8 m/s. Calculate the cyclist's kinetic energy. (Answer: 2240 J) KINETIC ENERGY IN REAL LIFE Applications: Vehicles: Heavier, faster vehicles have more kinetic energy, affecting impact force. Industrial Machinery: High-speed machinery holds significant kinetic energy. Compare the kinetic energy of a 2 kg book sliding at 5 m/s and a 0.5 kg ball moving at 10 m/s. (Answers: 25 J for the book, 25 J for the ball)  GRAV ITAT IONAL POTENT IAL ENERGY - UND ERSTAND ING AND FORMULA Eg=mgh, where g=9.8 m/s2 Example: A 48 kg person on a 110 m drop tower. Calculation: Eg=48×9.8×110=52 kJ Exercise: Calculate the potential energy of a 5 kg bag lifted 3 m above the ground. (Answer: 147 J) GRAVITATIONAL POTENTIAL ENERGY IN REAL LIFE Examples: Waterfalls: Water’s potential energy converts to kinetic as it falls. Construction Cranes: The height increases potential energy. What happens to the potential energy of a skier at the top of a slope as they descend? Describe the transformation. MECHANICAL ENERGY - COMBINING KINETIC AND POTENTIAL ENERGY Mechanical Energy Defined: The total energy of an object (kinetic + potential). Formula: Emechanical=Ek+Eg. Example: A roller coaster at the top of a hill combines both kinetic and potential energy. Calculate the total mechanical energy of a 50 kg skier at 10 m/s, standing 30 m above the ground. (Answer: Ek=2500J , Eg=14700J, total Emechanical=17200J) CONSERVATION OF MECHANICAL ENERGY Law of Conservation of Energy: Total mechanical energy in a closed system remains constant. Example: Pendulum: Energy shifts between kinetic and potential, but total remains the same. How energy transforms but remains conserved in a roller coaster ride. Identify points of maximum kinetic and potential energy. REAL-WORLD APPLICATION - ENERGY IN SUSTAINABLE TECHNOLOGIES Focus on Wind Power: Ontario’s Melancthon EcoPower Centre uses wind energy to power 52,000 homes. Exercise: List two advantages and two challenges of wind energy in comparison to fossil fuels. REAL-WORLD APPLICATION - OTHER SUSTAINABLE ENERGY SOURCES Renewable Energy Examples: Solar Panels: Convert sunlight to electricity. Geothermal: Uses Earth’s heat for energy. Biofuels: Organic material stores chemical potential energy. Research one form of renewable energy. Describe how it works and one limitation it faces in large-scale implementation.  RECAP OF KEY EQUATIONS Work:W=FΔd and W=F(cosθ)Δd. 1 Kinetic Energy: Ek= × 𝑚 × 𝑣 2. 2 Gravitational Potential Energy: Eg=mgh Mechanical Energy: Emechanical=Ek+Eg. Exercise: Identify which equation applies in these scenarios: Lifting a book. Pushing a cart. Dropping a ball. Moving a car. PRACTICE PROBLEMS Practice Problems 1.Work Problem: A person pulls a suitcase with 25 N force over 13 m. Find the work done. (Answer: 325 J) 2.Kinetic Energy Problem: Calculate the kinetic energy of a 70 kg runner moving at 10 m/s. (Answer: 3,500 J) 3.Potential Energy Problem: A 5 kg rock is 10 m above ground. Find Ep. (Answer: 490 J) REVIEW QUESTIONS Why does pulling at an angle reduce the effective work done? How does the conservation of energy principle apply to a swinging pendulum? Discuss benefits and limitations of using renewable energy sources like wind and solar power. CHAPTER 5.3 Types of Energy and the Law of Conservation of Energy Energy is a fundamental concept in physics, describing the ability to do work or cause change. In this section, we will explore the different types of energy, how they transform from one form to another, and the core principle of the Law of Conservation of Energy. How does energy move through systems, and why does it never "disappear"? TYPES OF ENERGY Energy exists in various forms, and each type plays a role in how work is done in different systems. Here’s a breakdown: Mechanical Energy: The sum of an object’s kinetic (movement) and potential (stored) energy. Example: A car speeding down a highway. Gravitational Energy: Energy due to position in a gravitational field. Example: A boulder at the top of a hill. Radiant Energy: Energy carried by electromagnetic waves (e.g., light). Example: Solar panels convert sunlight to electrical energy. Electrical Energy: Energy from moving electric charges. Example: Power from batteries or the grid.  Thermal Energy: Energy from the movement of particles in a substance (related to temperature). Example: Hot coffee cooling down. Sound Energy: Energy transferred through vibrations in a medium, such as air. Example: Music from a speaker. Chemical Energy: Energy stored in molecular bonds. Example: Gasoline combusting in an engine. Nuclear Energy: Energy stored in the nucleus of an atom. Example: Nuclear reactors splitting uranium atoms to release energy. Elastic Energy: Energy stored in stretched or compressed objects. Example: A stretched rubber band ready to snap. IDENTIFY ENERGY TYPES Look at the following examples and identify the type(s) of energy involved: A waterfall powering a hydroelectric dam. A toaster heating bread. A spring compressed in a mechanical clock. ENERGY TRANSFORMATIONS Energy transformations occur when energy changes from one form to another. This is a common process in nature and in engineered systems. In photosynthesis, plants convert radiant energy from sunlight into chemical energy stored in glucose. Some energy