One-Dimensional Motion PDF

General Physics 1  Example: Reference Frame A student is sitting inside a moving bus. Describe the A framework that is used for student’s motion with respect to the observation and mathematic...

General Physics 1  Example: Reference Frame A student is sitting inside a moving bus. Describe the A framework that is used for student’s motion with respect to the observation and mathematical the following frames of reference: description of physical phenomena and the formulation of physical 1. Bus laws. 2. Earth 3. Sun MOTION IS RELATIVE  An object is in motion if it changes position relative to a reference point. Objects that we call stationary—such as a tree, a sign, or a building— make good reference points. The passenger can use a tree as a reference point to decide if the train is moving. A tree makes a good reference point because it is stationary from the passenger’s point of view.  Describing Motion Whether or not an object is in motion depends on the reference point you choose.  All Motion is relative All motion is relative to a reference. This means that we describe the motion of an object relative to some other object. In our environment, the reference for motion is the earth’s surface, and the speeds are measured relative to the earth. The earth moves at 107,000 km/h relative to the sun Summary Motion is an object’s change in position relative to a reference point. The Earth’s surface is used as a common reference point A moving object can be used as a reference point as well  One- Dimensional Motion  KINEMATICS - the science of describing the motion of objects using words, diagrams, numbers, graphs, and equations Mechanics - the study of the motion of objects DYNAMICS - is the study of the motion of objects and the forces responsible for that motion  Kinematics language of kinematics used the following terms to describe a motion - distance and displacement - speed and velocity instantaneous and average - acceleration 4 Basic Quantities in Kinematics Concept Check 1. Distance is classified as which type of quantity? a. Vector b. Scalar c. Both vector and scalar d. Neither GENERAL PHYSICS 1 2. If Onzer moves 2 meters to the right and then 4 meters to the left, what is his total distance traveled? a. 2 meters b. 4 meters c. 6 meters d. 9 meters GENERAL PHYSICS 1 3. What does displacement indicate? a. The total distance traveled b. The overall change in position c. The speed of travel d. The time taken for the journey GENERAL PHYSICS 1 4. Why is understanding the difference between distance and displacement important in physics? a. It helps in calculating temperatures. b. It aids in understanding motion of objects. c. It is irrelevant to physics. d. It only applies to theoretical physics. GENERAL PHYSICS 1 5. If Onzer's initial position is 0 and he moves 3 meters left and then 5 meters right, what is his displacement? a. 2 meters to the right b. 5 meters to the right c. 3 meters to the left d. 8 meters to the right GENERAL PHYSICS 1 6. If the total distance traveled is 7 meters and the displacement is 5 meters, what can be inferred about the path taken? a. The path was straight b. The path included direction changes c. The path was circular d. The path was irrelevant GENERAL PHYSICS 1 7. What does the term "origin" refer to in the context of displacement? a. The initial position of an object b. The final position of an object c. The average position of an object d. The distance traveled GENERAL PHYSICS 1 8. In the context of the video, which statement is true? a. Distance can be negative. b. Displacement can be both positive and negative. c. Distance and displacement are the same. d. Displacement is not affected by direction. GENERAL PHYSICS 1 9. How does the direction of movement affect distance? a. It does not affect distance. b. It can make distance negative. c. It determines the total distance traveled. d. It only matters for displacement. GENERAL PHYSICS 1 10. What is the key difference between distance and displacement as mentioned in the video? a. Distance is always negative. b. Displacement requires direction; distance does not. c. Distance is a vector; displacement is a scalar. d. Both measure the same thing. GENERAL PHYSICS 1  Distance & Displacement  One Dimensional Position x Motion can be defined as the change of position over time. How can we represent position along a straight line? Position definition: Defines a starting point: origin (x = 0), x relative to origin Direction: positive (right or up), negative (left or down) It depends on time: t = 0 (start clock), x (t=0) does not have to be zero. Position has units of [Length]: meters. x = + 2.5 m x=-3m  Distance DISTANCE the sum of the actual path traveled A C B 10 m 5m d = 10 m + 5 m The fox moves from point A to point B then moves halfway back and stops at point C. d = 15 m Let’s determine the distance the fox has covered.  10 m 5m d = 10 m + 5 m Distance, being a scalar quantity of the actual path taken, is determined by simply d = 15 m adding the magnitudes of distance involved in the calculation. Distance  Displacement DISPLACEMENT the measurement of the change in position the distance from the starting point to the end point A C B 10 m 5m To the right To the left DISPLACEMENT the measurement of the change in position the distance from the starting point to the end point starting point/origin end point 5m A C B Displacement 10 m 5m Explanation: To the right To the left The change in position measures 5 m. Point C (end point) is 5 m to the right of Point A (starting point). 𝒅= 5 m to the right starting point/origin 5m end point Displacement A C B +y 10 m to the right 5 m to the left -x +x -y 𝒅 = 10 m + (-5 m) = 5m The usual sign convention used for 𝒅𝑹 = 5 m to the right vectors follows the Cartesian axes.  Distance & Displacement Distance ✓is a scalar quantity that refers to the total length of the path traveled between two positions. ✓No direction, thus, no sign Displacement ✓is a vector quantity that refers to "how far out of place an object is"; it is the object's overall change in position  Distance & Displacement EXAMPLE 1: A Physics teacher walks 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North. What is the total distance travelled by the physics teacher? How about the displacement? Total Distance: 12m Total Displacement: 0m The magnitude of the displacement is 0 because the teacher’s final position is same as the initial position  Distance & Displacement EXAMPLE 2 A football coach pacing back and forth along the sidelines. The diagram below shows several of coach's positions at various times. At each marked position, the coach makes a "U-turn" and moves in the opposite direction. In other words, the coach moves from position A to B to C to D. Total Distance: 95 yards Total Displacement: -55 yards The magnitude is 55 yards, and the direction is leftward  Distance & Displacement EXAMPLE 3 The diagram below shows the position of a cross-country skier at various times. At each of the indicated times, the skier turns around and reverses the direction of travel. In other words, the skier moves from A to B to C to D. Total Distance: 420 m Total Displacement: 140 m The magnitude is 140 meters, and the direction is rightward Concept Check 1. Distance is classified as which type of quantity? a. Vector b. Scalar c. Both vector and scalar d. Neither GENERAL PHYSICS 1 2. Which of the following describes velocity? a. A scalar quantity with magnitude only. b. A vector quantity with both magnitude and direction. c. Always a negative value. d. Independent of direction. GENERAL PHYSICS 1 3. What does the slope of a line on a position- time graph represent? a. The distance traveled b. The speed of the object c. The acceleration of the object d. The direction of motion GENERAL PHYSICS 1 4. How can we determine if an object is moving at a constant velocity on a position-time graph? a. The line is horizontal b. The line is vertical c. The line is a straight line d. The line is curved GENERAL PHYSICS 1 5. Which of the following is true about instantaneous velocity? a. It is the overall velocity during the entire journey. b. It is the velocity at a specific point in time. c. It does not have a direction. d. It is always constant. GENERAL PHYSICS 1 6. How can we determine instantaneous velocity from a graph? a. By finding the average speed. b. By drawing a tangent line at a point. c. By measuring the total distance. d. By counting the number of points on the graph. GENERAL PHYSICS 1  Speed, Velocity, & Acceleration  What is Speed? Speed is the distance an object travels in a certain amount of time. To calculate speed, you use the following formula: Speed (v) = Distance (d) Time (t) Standard Unit: meters per second (m/s) SPEED - the rate at which a body moves - distance traveled per unit time B d = 20 m 𝒅 𝟐𝟎 𝒎 V= = A 𝒕 𝟒𝒔 𝒗= 5 m/s Time t = 4 s  Ways To Calculate Speed ❑Constant Speed is when you are traveling at the same rate of speed, such as 55 mph constantly on a highway. ❑Average Speed is taking the total distance traveled, and dividing by the total time it takes. Used for calculations that involve changing speed. ❑Instantaneous Speed is the speed at any one given point in time.  Average Speed Bass Boat Speed What is the average speed of the bass boat depicted in the graph? 140 120 125 Average speed is taking 100 100 the total distance traveled (0 to 125 meters),and Distance (meters) 85 80 60 65 65 65 dividing by the total time 40 50 (1 to 9 seconds) it takes. 20 20 Average Speed = 125 meters = 15.6 m/s 8 seconds 0 0 1 2 3 4 5 6 7 8 9 10 Time (seconds)  Instantaneous Speed Bass Boat Speed What is the instantaneous speed of the bass boat at t=7 seconds? 140 Instantaneous speed is 125 120 100 100 speed at any given point in Distance (meters) 80 85 time. At 7 seconds, the 60 65 65 65 distance is 85 meters; 50 therefore 40 20 20 Instantaneous Speed = 85 meters = 12.1 m/s 0 0 1 2 3 4 5 6 7 8 9 10 7 seconds Time (seconds)  Speed Graphs Bass Boat Speed 70 1. In what time period is the bass boat speeding 65 65 65 60 50 50 up? Distance (meters) 45 40 2. In what time period is 30 the bass boat slowing 20 20 15 down? 10 3. When is the speed NOT 0 1 0 2 3 4 5 6 7 8 9 0 10 changing? Time (seconds)  Graphing Speed Speed is usually graphed Bass Boat Speed using a line graph, and it 70 depicts the distance and time. 65 65 65 60 Time is the independent 50 50 Distance (meters) 45 variable, and thus is ALWAYS 40 on the x-axis. 30 20 20 Distance is the dependent 15 10 variable, and thus is ALWAYS 0 0 0 on the y-axis. 1 2 3 4 5 6 7 8 9 10 Time (seconds)  The steepness of a line on a graph is called slope. The steeper the slope is, the greater the speed. A constant slope represents motion at constant speed. Using the points shown, the rise is 400 meters and the run is 2 minutes. To find the slope, you divide 400 meters by 2 minutes. The slope is 200 meters per minute.  What is Velocity? Velocity is the speed of an object, but the direction is also included. To calculate velocity, you use the following formula: Velocity (v) = Displacement (d) Time (t) Standard Unit: meters per second (m/s) VELOCITY - the speed of a moving body in a given direction 𝒅 𝟏𝟐 𝒎 d = 20 m B 𝒗=v = = = D 12 m 𝒕t 4s 𝟒 𝒔 𝒗 =12 m A 20o v = 3 m/s at 200 N of E Time t = 4 s Direction is required!  displacement x x f − xi Average Velocity vavg = = t t displacement Instantaneous x dx v = lim = t → 0  t Velocity dt What is Velocity? Average Velocity x x f − xi vavg = = t t Average velocity is the slope of the line segment between end points on a graph. Dimensions: length/time (L/T) [m/s]. SI unit: m/s. It is a vector (i.e. is signed), and displacement direction sets its sign.  Instantaneous Velocity Instantaneous means “at some given instant”. The instantaneous velocity indicates what is happening at every point of time. Limiting process: Chords approach the tangent as Δt => 0 Slope measure rate of change of position It is a vector quantity. Dimension: length/time (L/T), [m/s]. It is the slope of the tangent line to x(t). x dx v = lim = t → 0  t dt Instantaneous velocity v(t) is a function of time.  Uniform Velocity Uniform velocity is the special case of constant velocity In this case, instantaneous velocities are always the same, all the instantaneous velocities will also equal the average velocity x x − x Begin with v = t = t then x f = xi + v x t f i x Note: we are plotting x v velocity vs. time x(t) v(t) xf vx xi 0 t 0 t ti tf Jan. 28-Feb. 1, 2013  Graphical Interpretation of Velocity Velocity can be determined from a position-time graph Average velocity equals the slope of the line joining the initial and final positions. It is a vector quantity. An object moving with a constant velocity will have a graph that is a straight line. Jan. 28-Feb. 1, 2013  Checkpoint! A car moves 8 m in 4 s. Find (a) average speed, (b) instantaneous speed at t = 3 s. Time (s) 0 1 2 3 4 Distance (m) 0 2 4 6 8 0m 2m 4m 6m 8m  Checkpoint! 1. How are speed and velocity similar? They both measure how fast something is moving. 2. How are speed and velocity different? Velocity includes the direction of motion and speed does not (the car is moving 5mph East). 3. Is velocity more like distance or displacement? Why? Displacement, because it includes direction. Reminder… Do not forget to include your units. Concept Check 1. In physics, how is acceleration defined? a. The change in speed over distance b. The rate of change of velocity over time c. The total distance traveled over time d. The speed of an object at any point GENERAL PHYSICS 1 2. According to the video, what happens when the initial velocity and acceleration have opposite signs? a. The object is speeding up b. The object is slowing down c. The object is stationary d. The object's velocity is constant GENERAL PHYSICS 1 3. If a car has an initial velocity of 30 meters per second and comes to a complete stop in 2 seconds, what is its acceleration? a. -10 meters per second squared b. -15 meters per second squared c. 15 meters per second squared d. 30 meters per second squared GENERAL PHYSICS 1 4. What does the slope of a velocity versus time graph represent? a. Displacement b. Velocity c. Time d. Acceleration GENERAL PHYSICS 1 5. When a car is decelerating, what does a negative acceleration indicate? a. The car is moving backward b. The car is speeding up c. The car is slowing down d. The car is stationary GENERAL PHYSICS 1 6. How does changing direction affect acceleration? a. It does not affect acceleration b. It always increases acceleration c. It can cause acceleration even if speed remains constant d. It causes deceleration GENERAL PHYSICS 1 7. What is the formula for finding acceleration? a. Acceleration = final velocity - initial velocity b. Acceleration = (final velocity - initial velocity) / time c. Acceleration = speed / time d. Acceleration = distance / time² GENERAL PHYSICS 1 8. How can we find the acceleration of an object from a velocity versus time graph? a. By finding the area under the graph b. By finding the slope of the line c. By counting the time intervals d. By measuring the distance GENERAL PHYSICS 1  What is Acceleration? Acceleration is the rate of change in the velocity. To calculate acceleration, you use the following formula: 𝛥𝑣 v – v0 Acceleration (a) = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡𝑖𝑚𝑒 = 𝛥𝑡 = t – t0 Standard Unit: meters per second squared (m/s2 )  Speeding Up & Slowing Down If the acceleration is in the If the speed decreases, the same direction as the velocity, acceleration is in the opposite speed increases and direction from the velocity, then acceleration is positive acceleration is negative. An object accelerates when there’s a change in velocity. v v Car speeds up a v v Car slows up a  An object accelerates when there’s a change in its direction or it changes both velocity and direction. v a  Acceleration The acceleration is proportional to the magnitude of the force. The direction of acceleration is same as direction of force. Pulling the wagon with twice the force produces twice the acceleration and acceleration is in direction of force.  Acceleration & Velocity As velocity increases, so does acceleration As velocity decreases, so does acceleration When direction changes, so does acceleration When there is a constant velocity, there is no acceleration  Sample Problems AVERAGE SPEED Example 1 Every day, the teacher takes public transportation for 45 minutes to go to school 𝑑 about 20 km away from where she lives. 𝑣= What is the average speed? 𝑡 Given: d = 20 km 𝑡 = 45 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 Find: 𝑣 =? Given: d = 20 km 𝑑 𝑣= 𝑡 = 45 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 𝑡 20 𝑘𝑚 Find: 𝑣= 𝑣 =? 45 𝑚𝑖𝑛 𝒌𝒎 𝒗 = 𝟎. 𝟒𝟒 𝒎𝒊𝒏 AVERAGE SPEED Example 2 A car travels 20.0 m/s from home to the store and then travel 30.0 𝑣1 + 𝑣2 m/s from the store going to the 𝑣= 2 park. What is the average speed of the car for the entire trip? Given: Find: 𝑣1 = 20.0 𝑚/𝑠 𝑣𝑎𝑣𝑒𝑟𝑎𝑔𝑒 = ? 𝑣2 = 30.0 𝑚/𝑠 Given: 𝑣1 + 𝑣2 𝑣1 = 20.0 𝑚/𝑠 𝑣𝑎𝑣𝑒𝑟𝑎𝑔𝑒 = 𝑣2 = 30.0 𝑚/𝑠 2 20.0𝑚/𝑠 + 30.0 𝑚/𝑠 Find: 𝑣𝑎𝑣𝑒𝑟𝑎𝑔𝑒 = 𝑣𝑎𝑣𝑒𝑟𝑎𝑔𝑒 = ? 2 50.0𝑚/𝑠 𝑣𝑎𝑣𝑒𝑟𝑎𝑔𝑒 = 2 𝒗𝒂𝒗𝒆𝒓𝒂𝒈𝒆 = 𝟐𝟓. 𝟎 𝒎/𝒔 AVERAGE SPEED Example 3 Rachel watches a thunderstorm from her windows. She sees the flash of lightning bolt and begins 𝑑 counting the seconds until she hears the clap of 𝑣= thunder 5.00 seconds later. Assume that the speed of 𝑡 sound in air is 340 m/s and the lights was seen instantaneously. How far away was the lightning bolt? Given: 𝑡 = 5.00 𝑠 Find: 𝑣 = 340 𝑚/𝑠 𝑑 =?  𝑑 Given: 𝑡 𝑣= 𝑡 𝑡 𝑡 = 5.00 𝑠 𝑣 = 340 𝑚/𝑠 𝑣𝑡 = 𝑑 Find: 𝒅 = 𝒗𝒕 𝑑 =? 𝑚 𝑑 = 340 5.00 𝑠 𝑑 𝑠 𝑣= 𝑑 = 1700 𝑚 𝑡 𝒅 = 𝟏. 𝟕 𝒌𝒎 ACCELERATION Example 1 𝐹𝑂𝑅𝐶𝐸 𝑡 =3𝑠 𝑣റ𝑖 = 2 𝑚/𝑠 𝑣റ𝑓 = 8 𝑚/𝑠 A sailboat moving at the rate of 2 m/s is blown by the wind making it increase its speed to 8 m/s in 3 seconds. What is the acceleration of the boat? The boat increases its speed from 2m/s to 8 m/s in 3 s. Time (s) 0 1 2 3 Velocity (m/s) 2 4 6 8 2 m/s 4 m/s 6 m/s 8 m/s Given: 𝑣𝑓 − 𝑣0 ∆𝑣 𝑣𝑖 = 2 𝑚/𝑠 𝑎= = 𝑣𝑓 = 8 𝑚/𝑠 𝑡 𝑡 𝑡 =3𝑠 where: a = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 Find: 𝑣0 = 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 a=? 𝑣𝑓 = 𝑓𝑖𝑛𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑡 = 𝑡𝑖𝑚𝑒 Given: 𝑣𝑖 = 2 𝑚/𝑠 𝑣𝑓 − 𝑣𝑖 𝑣𝑓 = 8 𝑚/𝑠 𝑎= 𝑡 𝑡 =3𝑠 8 𝑚/𝑠 − 2 𝑚/𝑠 𝑎= 3𝑠 Find: 6 𝑚/𝑠 a=? 𝑎= 3𝑠 𝑚/𝑠 𝒂 = 𝟐 𝒎/𝒔𝟐 𝑎=2 𝑠  𝒂 = 𝟐 𝒎/𝒔𝟐 The wind changes the speed of a boat from 2 m/s to 8 m/s in 3 s. Each second the speed changes by 2 m/s. Wind force is constant and directed to the right, thus acceleration is constant and also directed to the right. ACCELERATION Example 2 Consider a moving particle to the right with motion details presented in the table below: Time (s) 0 1 2 3 4 Velocity (m/s) 0 2 4 6 8 Determine the average acceleration of the particle by dividing the change in velocity by the change in time used. ∆𝑣 𝑎= ∆𝑡  Time (s) 0 1 2 3 4 Given: Velocity 0 2 4 6 8 Find: (m/s) 𝑚 𝑚 𝑚 𝑚 𝑚 a=? 8 −6 −4 −2 −0 𝑎= 𝑠 𝑠 𝑠 𝑠 𝑠 4𝑠 −3𝑠 −2𝑠 −1𝑠 −0 ∆𝑣 𝑚 −4 𝑠 𝑎= 𝑎= ∆𝑡 −2 𝑠 𝑣𝑓 − 𝑣0 𝑚/𝑠 𝒂 = 𝟐 𝒎/𝒔𝟐 𝑎= 𝑎=2 𝑠 𝑡𝑓 − 𝑡0 From this, we conclude that the average acceleration of the particle is 𝒂 = 𝟐 𝒎/𝒔𝟐 to the right. Simple Word Problems  Speed, Velocity, Acceleration 1. What is the speed of a rocket that travels 9000 meters in 12.12 seconds? 2. What is the speed of a jet plane that travels 528 meters in 4 seconds? 3. How long will your trip take (in hours) if you travel 350 km at an average speed of 80 km/hr?  Speed, Velocity, Acceleration 4. How far (in meters) will you travel in 3 minutes running at a rate of 6 m/s? 5. A trip to Cape Canaveral, Florida takes 10 hours. The distance is 816 km. Calculate the average speed. 6. How many seconds will it take for a satellite to travel 450 km at a rate of 120 m/s?  Speed, Velocity, Acceleration 7. What is the speed of a walking person in m/s if the person travels 1000 m in 20 minutes? 8. A ball rolls down a ramp for 15 seconds. If the initial velocity of the ball was 0.8 m/s and the final velocity was 7 m/s, what was the acceleration of the ball ? 9. A meteoroid changed velocity from 1.0 km/s to 1.8 km/s in 0.03 seconds. What is the acceleration of the meteoroid?  Speed, Velocity, Acceleration 10. A car going 50mph accelerates to pass a truck. Five seconds later the car is going 80mph. Calculate the acceleration of the car. (1 mile = 1.61 km) 11. The space shuttle releases a space telescope into orbit around the earth. The telescope goes from being stationary to traveling at a speed of 1700 m/s in 25 seconds. What is the acceleration of the satellite? 12. A ball is rolled at a velocity of 12 m/s. After 36 seconds, it comes to a stop. What is the acceleration of the ball?  Speed, Velocity, Acceleration 13. How much force is needed to accelerate a truck with a mass of 2,000 kg, at a rate of 3 m/s²? 14. A dragster in a race accelerated from stop to 60 m/s by the time it reached the finish line. The dragster moved in a straight line and traveled from the starting line to the finish line in 8.0 s. What was the acceleration of the dragster? 15. A 300 N force acts on a 25 kg object. The acceleration of the object is ________. Challenge Problems  Challenge Problem Level 1 The position of an object moving along an x axis is given by x = 3t – 4t2 + t3, where x is in meters and t in seconds. Find the position of the object at the following values of t: (a)1s, (b) 2s, (c) 3 s, and (d) 4 s. (e) What is the object’s displacement between t = 0 and t = 4 s? (f) What is its average velocity for the time interval from t = 2 s to t = 4 s?  Challenge Problem Level 2 Two trains each having a speed of 30 km/h, are headed at each other on the same straight track. A bird that can fly 60 km/h flies off the front of one train when they are 60 km apart and heads directly for the other train. On reaching the other train, the crazy bird flies directly back to the first train, and so forth. What is the total distance the bird travels before the trains collide?  Challenge Problem Level 3 The position of a particle moving along the x axis is given in centimeters by x = 9.75 + 1.50t3 ,where t is in seconds. Calculate (a) the average velocity during the time interval t = 2.00 s to t = 3.00 s; (b) the instantaneous velocity at t = 2.00 s; (c) the instantaneous velocity at t = 3.00 s; (d) the instantaneous velocity at t = 2.50 s; and (e) the instantaneous velocity when the particle is midway between its positions at t = 2.00 s and t = 3.00 s.