Lecture 2: Motion in One Dimension PDF

LECTURE 2 CHAPTER 2. MOTION IN ONE DIMENSION 2.1 Position, Velocity, and Speed 2.2 Instantaneous Velocity and Speed 2.3 Analysis Model: Particle Under Constant Velocity 2.4 Acceleration 2.5 Motion Diagrams 2.6 Analysis Model: Particle Under Constant Acceleration 2.7 Freely Falling Objects ...

LECTURE 2 CHAPTER 2. MOTION IN ONE DIMENSION 2.1 Position, Velocity, and Speed 2.2 Instantaneous Velocity and Speed 2.3 Analysis Model: Particle Under Constant Velocity 2.4 Acceleration 2.5 Motion Diagrams 2.6 Analysis Model: Particle Under Constant Acceleration 2.7 Freely Falling Objects  Introduction: Dynamics ► The branch of physics involving the motion of an object and the relationship between that motion and other physics concepts ► Kinematics is a part of dynamics ▪ In kinematics, you are interested in the description of motion ▪ Not concerned with the cause of the motion ‣ Motion in one dimension: motion of an object along a straight line ‣ Particle model: describe the moving object as a particle regardless of its size (a particle is a point-like object)  2.1. Position and Displacement A ► Positionis defined in terms of a frame of reference Frame A: xi>0 and xf>0 y’ B Frame B: x’i0 ► One dimensional, so generally the x- or y-axis O’ x’ xi’ xf ’  2.1 Position and Displacement ► Positionis defined in terms of a frame of reference ▪ One dimensional, so generally the x- or y-axis ► Displacement measures the change in position ▪ Represented as x (if horizontal) or y (if vertical) Units ▪ Vector quantity SI Meters (m) ►+ or - is generally sufficient to indicate direction for one- CGS Centimeters (cm) dimensional motion US Cust Feet (ft)  Distance and Displacement ► Distance may be, but is not necessarily, the magnitude of the displacement Displacement Distance (red line) (blue line)  Position-time graphs Note: position-time graph is not necessarily a straight line, even though the motion is along x-direction  ConcepTest 1 An object (say, car) goes from one point in space to another. After it arrives to its destination, its displacement is 1. either greater than or equal to 2. always greater than 3. always equal to 4. either smaller or equal to 5. either smaller or larger than the distance it traveled.  ConcepTest 1 (answer) An object (say, car) goes from one point in space to another. After it arrives to its destination, its displacement is 1. either greater than or equal to 2. always greater than 3. always equal to 4. either smaller or equal to ✓ 5. either smaller or larger than the distance it traveled. Note: displacement is a vector from the final to initial points, distance is total path traversed  Speed ► Speed is a scalar quantity ▪ same units as velocity ▪ speed = total distance / total time ► May be, but is not necessarily, the magnitude of the velocity Example 2.1 Calculating the Average Velocity and Speed Find the displacement, average velocity, and average speed of the car in Figure 2.1a between positions A and F Figure 2.1 A car moves back and forth along a straight line. Because we are interested only in the car’s translational motion, we can model it as a particle. (a) A pictorial representation of the motion of the car. (b) (b) A graphical representation (position–time graph) of the motion of the car Example 2.1 Calculating the Average Velocity and Speed Find the displacement, average velocity, and average speed of the car in Figure 2.1a between positions A and F Position– time: The displacement of the car The car’s average velocity: The car’s average speed:  Uniform Velocity ► Uniform velocity is constant velocity ► The instantaneous velocities are always the same ▪ All the instantaneous velocities will also equal the average velocity Graphical Interpretation of Instantaneous Velocity ► Instantaneous velocity is the slope of the tangent to the curve at the time of interest ► The instantaneous speed is the magnitude of the instantaneous velocity  Average vs Instantaneous Velocity Average velocity Instantaneous velocity The average velocity : during a specific period of time Instantaneous velocity : slow and fast motion at time point  Average vs Instantaneous Velocity Average Velocity Instantaneous Velocity  ConcepTest 2 The graph shows position as a function of time for two trains running on parallel tracks. Which of the following is true: 1. at time tB both trains have the same velocity 2. both trains speed up all the time 3. both trains have the same velocity at some time before tB 4. train A is longer than train B 5. all of the above statements are true position A B tB time  ConcepTest 2 (answer) The graph shows position as a function of time for two trains running on parallel tracks. Which of the following is true: 1. at time tB both trains have the same velocity 2. both trains speed up all the time 3. both trains have the same velocity at some time before tB 4. train A is longer than train B 5. all of the above statements are true position A B Note: the slope of curve B is parallel to line A at some point t< tB tB time Example 2.2 The Velocity of Different Objects Consider the following one-dimensional motions: (A) a ball thrown directly upward rises to a highest point and falls back into the thrower’s hand; (B) a race car starts from rest and speeds up to 100 m/s (C) a spacecraft drifts through space at constant velocity. Are there any points in the motion of these objects at which the instantaneous velocity has the same value as the average velocity over the entire motion? If so, identify the point(s). Example 2.2 The Velocity of Different Objects Consider the following one-dimensional motions: (A) a ball thrown directly upward rises to a highest point and falls back into the thrower’s hand; (B) a race car starts from rest and speeds up to 100 m/s (C) a spacecraft drifts through space at constant velocity. Are there any points in the motion of these objects at which the instantaneous velocity has the same value as the average velocity over the entire motion? If so, identify the point(s). (A) Because the ball returns to the starting point: vaver (for the thrown ball) = 0 → x = 0 → There is one point at which the instantaneous velocity is zero: at the top of the motion. (B) vaver (of the car) cannot be evaluated unambiguously with the information given, but vaver = 0 - 100 m/s. Because every vinst = 0 and 100 m/s at some time during the interval → there must be some instant at which the vinst is equal to the vaver over the entire motion. (C) vinst (the spacecraft) = constant → vinst at any time and its vaver over any time interval are the same.  Example 2.3 Average and Instantaneous Velocity A particle moves along the x axis. Its position varies with time according to the expression x = -4t + 2t2, where x is in meters and t is in seconds. The position– time graph for this motion is shown in Figure 2.4a. Because the position of the particle is given by a mathematical function, the motion of the particle is completely known, unlike that of the car in Figure 2.1. Notice that the particle moves in the negative x direction for the first second of motion, is momentarily at rest at the moment t = 1s, and moves in the positive x direction at times t > 1 s. (A) Determine the displacement of the particle in the time intervals t = 0 to t = 1s and t = 1s to t = 3s (B) Calculate the average velocity during these two time intervals (C) Find the instantaneous velocity of the particle at t = 2.5 s.  Example 2.3 Average and Instantaneous Velocity (A) Determine the displacement of the particle in the time intervals t = 0 to t = 1s and t = 1s to t = 3s (B) Calculate the average velocity during these two time intervals (C) Find the instantaneous velocity of the particle at t = 2.5 s.  2.2. Analysis Model: Particle Under Constant Velocity Analysis model: It is a common situation that occurs time and again when solving physics problems All of the analysis models that we will develop are based on four fundamental simplification models: particle, system, a rigid object, and a wave Examples: ▪ a planet traveling around a perfectly circular orbit ▪ a car traveling at a constant speed on a curved racetrack ▪ a runner traveling at constant speed on a curved path ▪ a charged particle moving through a uniform magnetic field Two basic steps to solve a problem: Identify the analysis model that is appropriate for the problem The model tells you which equation(s) to use for the mathematical representation  2.2. Analysis Model: Particle Under Constant Velocity Analysis model: particle under constant velocity If the velocity of a particle is constant, its instantaneous velocity at any instant during a time interval is the same as the average velocity over the interval ư the position of the particle as a function of time given by  Example 2.4 Modeling a Runner as a Particle A kinesiologist is studying the biomechanics of the human body. (Kinesiology is the study of the movement of the human body. Notice the connection to the word kinematics.) She determines the velocity of an experimental subject while he runs along a straight line at a constant rate. The kinesiologist starts the stopwatch at the moment the runner passes a given point and stops it after the runner has passed another point 20 m away. The time interval indicated on the stopwatch is 4.0 s. (A) What is the runner’s velocity? (B) If the runner continues his motion after the stopwatch is stopped, what is his position after 10 s have passed?  Example 2.4 Modeling a Runner as a Particle A kinesiologist is studying the biomechanics of the human body. (Kinesiology is the study of the movement of the human body. Notice the connection to the word kinematics.) She determines the velocity of an experimental subject while he runs along a straight line at a constant rate. The kinesiologist starts the stopwatch at the moment the runner passes a given point and stops it after the runner has passed another point 20 m away. The time interval indicated on the stopwatch is 4.0 s. (A) What is the runner’s velocity? (B) If the runner continues his motion after the stopwatch is stopped, what is his position after 10 s have passed?  2.4. Average Acceleration ► When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is increasing ► When the sign of the velocity and the acceleration are opposite, the speed is decreasing Units SI Meters per second squared (m/s2) CGS Centimeters per second squared (cm/s2) US Customary Feet per second squared (ft/s2)  Graphical Interpretation of Acceleration ► Average acceleration is the slope of the line connecting the initial and final velocities on a velocity-time graph ► Instantaneous acceleration is the slope of the tangent to the curve of the velocity- time graph  Example 2.6 Average and Instantaneous Acceleration The velocity of a particle moving along the x axis varies according to the expression vx = 40 - 5t2, where vx is in meters per second and t is in seconds. (A) Find the average acceleration in the time interval t = 0 to t= 2.0s (B) Determine the acceleration at t = 2.0s.  Example 2.6 Average and Instantaneous Acceleration The velocity of a particle moving along the x axis varies according to the expression vx = 40 - 5t2, where vx is in meters per second and t is in seconds. (A) Find the average acceleration in the time interval t = 0 to t = 2.0 s (B) Determine the acceleration at t = 2.0 s.  2.5. Motion Diagrams A motion diagram is sometimes useful to describe the velocity and acceleration while an object is in motion ► Uniform velocity (shown by red arrows maintaining the same size) ► Acceleration equals zero  Example ► Velocity and acceleration are in the same direction ► Acceleration is uniform (blue arrows maintain the same length) ► Velocity is increasing (red arrows are getting longer)  Example ► Acceleration and velocity are in opposite directions ► Acceleration is uniform (blue arrows maintain the same length) ► Velocity is decreasing (red arrows are getting shorter)  2.6. One-dimensional Motion With Constant Acceleration ► If acceleration is uniform (i.e. a = a ): v f − vo v f − vo a= = thus: t f − t0 t v f = vo + at ◼ Shows velocity as a function of acceleration and time  2.6. One-dimensional Motion With Constant Acceleration ► Used in situations with uniform acceleration v f = vo + at  vo + v f   x = v av eraget =   t  2  1 2  x = v o t + at Velocity changes uniformly!!! 2 v = v + 2ax 2 f 2 o  Notes on the equations  vo + v f   x = v average t= t  2  ► Gives displacement as a function of velocity and time 1 2  x = v o t + at 2 ► Gives displacement as a function of time v = v + 2ax 2 f 2 o ► Gives velocity as a function of displacement  Example 2.7 Carrier Landing A jet lands on an aircraft carrier at a speed of 140 mi/h (< 63 m/s). (A) What is its acceleration (assumed constant) if it stops in 2.0 s due to an arresting cable that snags the jet and brings it to a stop? (B) If the jet touches down at position xi = 0, what is its final position?  Example 2.7 Carrier Landing A jet lands on an aircraft carrier at a speed of 140 mi/h (< 63 m/s). (A) What is its acceleration (assumed constant) if it stops in 2.0 s due to an arresting cable that snags the jet and brings it to a stop? (B) If the jet touches down at position xi = 0, what is its final position?  Example 2.8 Watch Out for the Speed Limit A car traveling at a constant speed of 45.0 m/s passes a trooper on a motorcycle hidden behind a billboard. One second after the speeding car passes the billboard, the trooper sets out from the billboard to catch the car, accelerating at a constant rate of 3.00 m/s2. How long does it take the trooper to overtake the car?  Example 2.8 Watch Out for the Speed Limit A car traveling at a constant speed of 45.0 m/s passes a trooper on a motorcycle hidden behind a billboard. One second after the speeding car passes the billboard, the trooper sets out from the billboard to catch the car, accelerating at a constant rate of 3.00 m/s2. How long does it take the trooper to overtake the car? Summary of kinematic equations Exercises See a clip  2.7 Free Fall ► All objects moving under the influence of only gravity are said to be in free fall ► All objects falling near the earth’s surface fall with a constant acceleration ► This acceleration is called the acceleration due to gravity, and indicated by g  Acceleration due to Gravity ► Symbolized by g ► g = 9.8 m/s² (can use g = 10 m/s² for estimates) ► g is always directed downward ▪ toward the center of the earth Free Fall -- an Object Dropped y ► Initial velocity is zero ► Frame: let up be positive ► Use the kinematic equations x ▪ Generally use y instead vo= 0 of x since vertical a=g 1 2 y = at 2 a = − 9.8 m s 2  Free Fall -- an Object Thrown Downward ►a =g ▪ With upward being positive, acceleration will be negative, g = -9.8 m/s² ► Initial velocity  0 ▪ With upward being positive, initial velocity will be negative Free Fall -- object thrown upward ► Initialvelocity is upward, so positive v=0 ► The instantaneous velocity at the maximum height is zero ► a = g everywhere in the motion ▪ g is always downward, negative  Thrown upward ► The motion may be symmetrical ▪ then tup = tdown ▪ then vf = -vo ► The motion may not be symmetrical ▪ Break the motion into various parts ►generally up and down  Non-symmetrical Free Fall ► Need to divide the motion into segments ► Possibilities include ▪ Upward and downward portions ▪ The symmetrical portion back to the release point and then the non- symmetrical portion  Combination Motions rocket fly up and then fall the ground  ConcepTest 3 A person standing at the edge of a cliff throws one ball straight up and another ball straight down at the same initial speed. Neglecting air resistance, the ball to hit ground below the cliff with greater speed is the one initially thrown 1. upward 2. downward 3. neither – they both hit at the same speed  ConcepTest 3 (answer) A person standing at the edge of a cliff throws one ball straight up and another ball straight down at the same initial speed. Neglecting air resistance, the ball to hit ground below the cliff with greater speed is the one initially thrown 1. upward 2. downward 3. neither – they both hit at the same speed Note: upon the descent, the velocity of an object thrown straight up with an initial velocity v is exactly –v when it passes the point at which it was first released.  Example 2.10 Not a Bad Throw for a Rookie A stone thrown from the top of a building is given an initial velocity of 20.0 m/s straight upward. The stone is launched 50.0 m above the ground, and the stone just misses the edge of the roof on its way down as shown in Figure 2.14. (A) Using tA = 0 as the time the stone leaves the thrower’s hand at position A, determine the time at which the stone reaches its maximum height. (B) Find the maximum height of the stone. (C) Determine the velocity of the stone when it returns to the height from which it was thrown (D) Find the velocity and position of the stone at t = 5.00 s  Example 2.10 Not a Bad Throw for a Rookie A stone thrown from the top of a building is given an initial velocity of 20.0 m/s straight upward. The stone is launched 50.0 m above the ground, and the stone just misses the edge of the roof on its way down as shown in Figure 2.14. (A) Using tA = 0 as the time the stone leaves the thrower’s hand at position A, determine the time at which the stone reaches its maximum height. (B) Find the maximum height of the stone.  Example 2.10 Not a Bad Throw for a Rookie A stone thrown from the top of a building is given an initial velocity of 20.0 m/s straight upward. The stone is launched 50.0 m above the ground, and the stone just misses the edge of the roof on its way down as shown in Figure 2.14. (C) Determine the velocity of the stone when it returns to the height from which it was thrown. (D) Find the velocity and position of the stone at t = 5.00 s