1 D Kinematics Kinematics is the descriptive study of motion in which we attempt to simply describe motion: how fast, how far, how long, etc.?

One Dimensional Motion
3 Assumptions: Motion is confined to 1 dimension. Forces that cause motion are not specified. Object is a point or particle. A particle is either: A point-like object (such as an electron) Or an object that moves such that each part travels in the same direction at the same rate (no rotation or stretching) To simplify the concept of motion, we will first consider motion that takes place in one dimension. One example is the motion of a commuter train on a straight track. The train can either move forward or backward along a straight line. To describe motion, you must first choose a frame of reference. A frame of reference is a system for specifying the precise location of objects in space and time. The frame of reference for one dimensional motion can be thought of as a number line. Without a frame of reference, it would be impossible to meaningfully describe the motion of any object.

Position and Displacement
Position is measured relative to a reference point: The origin, or zero point, of an axis Position has a sign: Positive direction is in the direction of increasing numbers Negative direction is opposite the positive

A change in position is called displacement ∆x is the change in x, (final position) – (initial position) Eq. (2-1) Examples A particle moves . . . From x = 5 m to x = 12 m: ∆x = 7 m (positive direction) From x = 5 m to x = 1 m: ∆x = -4 m (negative direction) From x = 5 m to x = 200 m to x = 5 m: ∆x = 0 m The actual distance covered is irrelevant

Displacement Concept Check
Does the displacement of an object depend on the specific location of the origin of the coordinate system? Yes No It depends on the coordinate system 10 20 30 40 50 10 20 30 40 50 30 40 50 60 70 30 40 50 60 70

Displacement Concept Check
Does the displacement of an object depend on the specific location of the origin of the coordinate system? Yes No It depends on the coordinate system 10 20 30 40 50 10 20 30 40 50 30 40 50 60 70 30 40 50 60 70 Since the displacement is the difference between two coordinates, the origin does not matter.

Displacement is therefore a vector quantity Direction: along a single axis, given by sign (+ or -) Magnitude: length or distance, in this case meters or feet Ignoring sign, we get its magnitude (absolute value) The magnitude of ∆x = -4 m is 4 m. Answer: pairs (b) and (c) (b) -7 m – -3 m = -4 m (c) -3 m – 7 m = -10 m

Average Velocity Average velocity is the ratio of: A displacement, ∆x
To the time interval in which the displacement occurred, ∆t Average velocity has units of (distance) / (time) Meters per second, m/s

Average Velocity On a graph of x vs. t, the average velocity is the slope of the straight line that connects two points Average velocity is a vector quantity Positive slope means positive average velocity Negative slope means negative average velocity

Average Speed Average speed is the ratio of:
The total distance covered To the time interval in which the distance was covered, ∆t Average speed is always positive (scalar) Example A particle moves from x = 3 m to x = -3 m in 2 seconds. Average velocity = -3 m/s; average speed = 3 m/s

Position Graph Concept Check
The graph of position versus time for a car is given below. What can you say about the velocity of the car over time? t x 1. it speeds up all the time 2. it slows down all the time 3. it moves at constant velocity 4. sometimes it speeds up and sometimes it slows down 5. not really sure

The graph of position versus time for a car is given below. What can you say about the velocity of the car over time? t x 1. it speeds up all the time 2. it slows down all the time 3. it moves at constant velocity 4. sometimes it speeds up and sometimes it slows down 5. not really sure The car moves at a constant velocity because the x vs. t plot shows a straight line. The slope of a straight line is constant. Remember that the slope of x versus t is the velocity.

The graph of position versus time for a car is given below. What can you say about the velocity of the car over time? t x 1. it speeds up all the time 2. it slows down all the time 3. it moves at constant velocity 4. sometimes it speeds up and sometimes it slows down 5. not really sure

The graph of position versus time for a car is given below. What can you say about the velocity of the car over time? 1. it speeds up all the time 2. it slows down all the time 3. it moves at constant velocity 4. sometimes it speeds up and sometimes it slows down 5. not really sure The car slows down all the time because the slope of the x vs. t graph is diminishing as time goes on. Remember that the slope of x vs. t is the velocity. At large t, the value of the position x does not change, indicating that the car must be at rest.

Instantaneous Velocity
Instantaneous velocity, or just velocity, v, is: At a single moment in time Obtained from average velocity by shrinking ∆t View animation The slope of the position-time curve for a particle at an instant (the derivative of position) A vector quantity with units (distance) / (time) The sign of the velocity represents its direction Velocity is the derivative of x (dx) with respect to t (dt) Velocity is the rate at which the position is changing at a given instant.

Secant Approaching the Tanget

Example 2.02 Example The graph shows the position and velocity of an elevator cab over time. The slope of x(t), and so also the velocity v, is zero from 0 to 1s, and from 9s on. During the interval bc, the slope is constant and nonzero, so the cab moves with constant velocity (4 m/s).

Sample 2.02 The graph shows the velocity and acceleration of an elevator cab over time. When acceleration is 0 (e.g. interval bc) velocity is constant. When acceleration is positive (ab) upward velocity increases. When acceleration is negative (cd) upward velocity decreases. Steeper slope of the velocity-time graph indicates a larger magnitude of acceleration: the cab stops in half the time it takes to get up to speed.

GUESS Method Given Unknown Equation(s) Substitute Significant Figures
State the relevant given information in variable form. Include a diagram when appropriate. Unknown State the unknown variable (that which you are attempting to find). Equation(s) State the appropriate equation(s). Solve for the unknown if not already. Substitute Substitute the given information into the appropriate equation(s). Include units with all values. Significant Figures Round the answer to the appropriate number of significant figures based on the given information. Solve the problem on the template provided, following the GUESS Method.

Acceleration The rate of change in a particle's velocity is acceleration Average acceleration over a time interval ∆t is Instantaneous acceleration (or just acceleration), a, for a single moment in time is: Slope of velocity vs. time graph

Acceleration Acceleration is a vector quantity:
Positive sign means velocity is increasing Negative sign means the opposite Units of (distance / time / time)

Acceleration Example If a car with velocity vi = -25 m/s is braked to a stop (vf = 0 m/s) in 5.0 s, then a = m/s2. Acceleration is positive, because velocity increased. Note: accelerations can be expressed in units of g Answers: (a) + (b) - (c) - (d) +

Sample Problem 2.03 Find v(t) and a(t)
If a car is driving along a road and is at position xi at some time and position xf at some later time, the average velocity of the car can be determined by the change in position divided by the time it took to make that change. Of course, if the car turned around and traveled in the opposite direction, the change in position (displacement) would be negative, and so would the average velocity.

Velocity Graph Concept Check
Consider the line labeled A in the v versus t plot. How does the speed change with time for line A? 1. decreases 2. increases 3. stays constant 4. increases, then decreases 5. decreases, then increases v t A

Consider the line labeled A in the v versus t plot. How does the speed change with time for line A? 1. decreases 2. increases 3. stays constant 4. increases, then decreases 5. decreases, then increases v t A In case A, the initial velocity is positive and the magnitude of the velocity continues to increase with time.

Consider the line labeled A in the v versus t plot. How does the speed change with time for line A? 1. decreases 2. increases 3. stays constant 4. increases, then decreases 5. decreases, then increases v t A

Consider the line labeled B in the v versus t plot. How does the speed change with time for line B? v t A B 1. decreases 2. increases 3. stays constant 4. increases, then decreases 5. decreases, then increases

Consider the line labeled B in the v versus t plot. How does the speed change with time for line B? v t A B 1. decreases 2. increases 3. stays constant 4. increases, then decreases 5. decreases, then increases In case B, the initial velocity is positive and the magnitude of the velocity decreases toward zero. This will bring the object to rest briefly. Then the velocity becomes negative which means the object has changed direction. Velocity becomes more negative which means it speeds up in the opposite direction.

Constant Acceleration
In many cases acceleration is constant, or nearly so. For these cases, 5 special equations can be used. Note that constant acceleration means a velocity with a constant slope, and a position with varying slope (unless a = 0). Figure 2-9

Slope varies Slope = a Slope = 0

                        Kinematics Equations x
xo v vo a t                        

Sample Problem 2.04

Table 2-1 shows the 5 equations and the quantities missing from them. Answer: Situations 1 (a = 0) and 4.

Acceleration as Integration

Law of Falling Bodies

Hammer and Feather

Hammer and Feather (2)

Free-Fall Acceleration
Free-fall acceleration is the rate at which an object accelerates downward in the absence of air resistance Varies with latitude and elevation Written as g, standard value of 9.8 m/s2 Independent of the properties of the object (mass, density, shape, see Figure 2-12) The equations of motion in Table 2-1 apply to objects in free-fall near Earth's surface In vertical flight (along the y axis) Where air resistance can be neglected

Free-Fall Acceleration
The free-fall acceleration is downward (-y direction) Value ay = -g in the constant acceleration equations Answers: (a) The sign is positive (the ball moves upward); (b) The sign is negative (the ball moves downward); (c) The ball's acceleration is always m/s2 at all points along its trajectory

Sample Problem 2.05

1 v t 3 v t 4 v t 2 You drop a rubber ball. Right after it leaves your hand and before it hits the floor, which of the above plots represents the v vs. t graph for this motion? (Assume your y-axis is pointing up.)

1 v t 3 v t 4 v t 2 You drop a rubber ball. Right after it leaves your hand and before it hits the floor, which of the above plots represents the v vs. t graph for this motion? (Assume your y-axis is pointing up.) The ball is dropped from rest, so its initial velocity is zero. Since the y-axis is pointing upwards and the ball is falling downwards, its velocity is negative and becomes more and more negative as it accelerates downward.

Velocity Graph Concept Check (2)
1 1 v t 2 v t 3 v t 4 4 2 3 You toss a ball straight up in the air and catch it again. Right after it leaves your hand and before you catch it, which of the above plots represents the v vs. t graph for this motion? (Assume the y-axis is pointing up.)

1 1 v t 2 v t 3 v t 4 4 2 3 You toss a ball straight up in the air and catch it again. Right after it leaves your hand and before you catch it, which of the above plots represents the v vs. t graph for this motion? (Assume the y-axis is pointing up.) The ball has an initial velocity that is positive but diminishing as it slows. It stops at the top (v = 0), and then its velocity becomes negative and becomes more and more negative as it accelerates downward.

Velocity and Acceleration of an Object in Free Fall
Position Time Velocity Time Describe the motion of the object whose velocity is as shown. Time Acceleration -9.81 m/s2

Concept Check – Free Fall
You throw a ball straight up into the air. After it leaves your hand, at what point in its flight does it have the maximum value of acceleration? 1. its acceleration is constant everywhere 2. at the top of its trajectory 3. halfway to the top of its trajectory 4. just after it leaves your hand 5. just before it returns to your hand on the way down

You throw a ball straight up into the air. After it leaves your hand, at what point in its flight does it have the maximum value of acceleration? 1. its acceleration is constant everywhere 2. at the top of its trajectory 3. halfway to the top of its trajectory 4. just after it leaves your hand 5. just before it returns to your hand on the way down The ball is in free fall once it is released. Therefore, it is entirely under the influence of gravity, and the only acceleration it experiences is g, which is constant at all points.

Alice and Bill are at the top of a building. Alice throws her ball downward. Bill simply drops his ball. Which ball has the greater acceleration just after release? 1. Alice’s ball 2. it depends on how hard the ball was thrown 3. Neither – they both have the same acceleration 4. Bill’s ball v0 Bill Alice vA vB

Alice and Bill are at the top of a building. Alice throws her ball downward. Bill simply drops his ball. Which ball has the greater acceleration just after release? 1. Alice’s ball 2. it depends on how hard the ball was thrown 3. Neither – they both have the same acceleration 4. Bill’s ball Both balls are in free fall once they are released, therefore they both feel the acceleration due to gravity (g). This acceleration is independent of the initial velocity of the ball. v0 Bill Alice vA vB Follow-up: Which one has the greater velocity when they hit the ground?

You throw a ball upward with an initial speed of 10 m/s. Assuming that there is no air resistance, what is its speed when it returns to you? 1. more than 10 m/s 2. 10 m/s 3. less than 10 m/s 4. zero 5. need more information

You throw a ball upward with an initial speed of 10 m/s. Assuming that there is no air resistance, what is its speed when it returns to you? 1. more than 10 m/s 2. 10 m/s 3. less than 10 m/s 4. zero 5. need more information The ball is slowing down on the way up due to gravity. Eventually it stops. Since a = -g on the way up AND on the way down, the ball reaches the same speed when it gets back to you as it had when it left.

Alice and Bill are at the top of a cliff of height H. Both throw a ball with initial speed v0, Alice straight down and Bill straight up. The speeds of the balls when they hit the ground are vA and vB. If there is no air resistance, which is true? 1. vA < vB 2. vA = vB 3. vA > vB 4. impossible to tell v0 Bill Alice H vA vB v0 Bill Alice H vA vB [CORRECT 5 ANSWER]

Alice and Bill are at the top of a cliff of height H. Both throw a ball with initial speed v0, Alice straight down and Bill straight up. The speeds of the balls when they hit the ground are vA and vB. If there is no air resistance, which is true? 1. vA < vB 2. vA = vB 3. vA > vB 4. impossible to tell v0 Bill Alice H vA vB Bill’s ball goes up and comes back down to Bill’s level. At that point, it is moving downward with v0, the same as Alice’s ball. Thus, it will hit the ground with the same speed as Alice’s ball. [CORRECT 5 ANSWER] Follow-up: What happens if there is air resistance?

You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their velocities? 1. They both increase at the same rate 2. The velocity of the first rock increases faster than the second 3. The velocity of the second rock increases faster than the first 4. Both velocities remain constant

You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their velocities? 1. They both increase at the same rate 2. The velocity of the first rock increases faster than the second 3. The velocity of the second rock increases faster than the first 4. Both velocities remain constant Both rocks are in free fall, thus under the influence of gravity only. That means they both experience the constant acceleration of gravity. Since acceleration is defined as the change of velocity, both of their velocities increase at the same rate. Follow-up: What happens when air resistance is present?

You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their separation? 1. the separation increases as they fall 2. the separation stays constant at 4 m 3. the separation decreases as they fall 4. it is impossible to answer without more information

You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their separation? 1. the separation increases as they fall 2. the separation stays constant at 4 m 3. the separation decreases as they fall 4. it is impossible to answer without more information At any given time, the first rock always has a greater velocity than the second rock, therefore it will always be increasing its lead as it falls. Thus, the separation will increase.

Free Fall Concept Check
When throwing a ball straight up, which of the following is true about its velocity v and its acceleration a at the highest point in its path? 1. both v = 0 and a = 0 2. v ¹ 0, but a = 0 3. v = 0, but a ¹ 0 4. both v ¹ 0 and a ¹ 0 5. not really sure

Free Fall Concept Check
When throwing a ball straight up, which of the following is true about its velocity v and its acceleration a at the highest point in its path? 1. both v = 0 and a = 0 2. v ¹ 0, but a = 0 3. v = 0, but a ¹ 0 4. both v ¹ 0 and a ¹ 0 5. not really sure At the top, clearly v = 0 because the ball has momentarily stopped. But the velocity of the ball is changing, so its acceleration is definitely not zero. Otherwise it would remain at rest! Follow-up: …and the value of a is…?

1 v t 3 v t 4 v t 2 You drop a rubber ball. Right after it leaves your hand and before it hits the floor, which of the above plots represents the v vs. t graph for this motion? (Assume the y-axis is pointing up.)

1 v t 3 v t 4 v t 2 You drop a rubber ball. Right after it leaves your hand and before it hits the floor, which of the above plots represents the v vs. t graph for this motion? (Assume the y-axis is pointing up.) Initially, the ball is falling down, so its velocity must be negative (if UP is positive). Its velocity is also increasing in magnitude as it falls. Once it bounces, it changes direction and then has a positive velocity, which is also decreasing as the ball moves upward.

Graphical Integration in Motion Analysis
Learning Objectives 2.18 Determine a particle's change in velocity by graphical integration on a graph of acceleration versus time. 2.19 Determine a particle's change in position by graphical integration on a graph of velocity versus time.

Integrating acceleration: Given a graph of an object's acceleration a versus time t, we can integrate to find velocity The Fundamental Theorem of Calculus gives: Eq. (2-27) The definite integral on the right can be evaluated from a graph: Eq. (2-28)

Integrating velocity: Given a graph of an object's velocity v versus time t, we can integrate to find position The Fundamental Theorem of Calculus gives: Eq. (2-29) The definite integral on the right can be evaluated from a graph: Eq. (2-30)

Sample Problem 2.06 The graph shows the acceleration of a person's head and torso in a whiplash incident. To calculate the torso speed at t = s (assuming an initial speed of 0), find the area under the pink curve: area A = 0 area B = 0.5 (0.060 s) (50 m/s2) = 1.5 m/s area C = (0.010 s) (50 m/s2) = 0.50 m/s total area = 2.0 m/s

Acceleration Slope (Derivative) Area (Integral) Slope (Derivative)
Position Slope (Derivative) Area (Integral) Time Velocity Time Slope (Derivative) Area (Integral) Describe the motion of the object whose velocity is as shown. Time Acceleration

Summary of Ch. 2 Position Displacement Average Velocity Average Speed
Relative to origin Positive and negative directions Displacement Change in position (vector) Eq. (2-1) Average Velocity Displacement / time (vector) Average Speed Distance traveled / time Eq. (2-2) Eq. (2-3)

Summary of Ch. 2 Instantaneous Velocity Average Acceleration
At a moment in time Speed is its magnitude Average Acceleration Ratio of change in velocity to change in time Eq. (2-7) Eq. (2-4) Instantaneous Acceleration First derivative of velocity Second derivative of position Constant Acceleration Includes free-fall, where a = -g along the vertical axis Eq. (2-8) Tab. (2-1)

1 D Kinematics Kinematics is the descriptive study of motion in which we attempt to simply describe motion: how fast, how far, how long, etc.?

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1 D Kinematics Kinematics is the descriptive study of motion in which we attempt to simply describe motion: how fast, how far, how long, etc.?

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