Kinematics in One Dimension Chapter 2 Kinematics in One Dimension

Kinematics deals with the concepts that are needed to describe motion. Dynamics deals with the effect that forces have on motion. Together, kinematics and dynamics form the branch of physics known as Mechanics.

2.1 Displacement

2.1 Displacement

2.1 Displacement

2.1 Displacement

2.2.1. The branch of physics that deals with motion is called mechanics. Kinematics is the portion of mechanics that describes motion without any reference to which of the following concepts? a) forces b) accelerations c) velocities d) displacements e) time

2.2.1. The branch of physics that deals with motion is called mechanics. Kinematics is the portion of mechanics that describes motion without any reference to which of the following concepts? a) forces b) accelerations c) velocities d) displacements e) time

2.1.2. A particle travels along a curved path between two points A and B as shown. Complete the following statement: The displacement of the particle does not depend on a) the location of A. b) the location of B. c) the direction of A from B. d) the distance traveled from A to B. e) the shortest distance between A and B.

2.1.2. A particle travels along a curved path between two points A and B as shown. Complete the following statement: The displacement of the particle does not depend on a) the location of A. b) the location of B. c) the direction of A from B. d) the distance traveled from A to B. e) the shortest distance between A and B.

2.1.3. For which one of the following situations will the path length equal the magnitude of the displacement? a) An Olympic athlete is running around an oval track. b) A roller coaster car travels up and down two hills. c) A truck travels 4 miles west; and then, it stops and travels 2 miles west. d) A ball rises and falls after being thrown straight up from the earth's surface. e) A ball on the end of a string is moving in a vertical circle.

2.1.3. For which one of the following situations will the path length equal the magnitude of the displacement? a) An Olympic athlete is running around an oval track. b) A roller coaster car travels up and down two hills. c) A truck travels 4 miles west; and then, it stops and travels 2 miles west. d) A ball rises and falls after being thrown straight up from the earth's surface. e) A ball on the end of a string is moving in a vertical circle.

2.1.4. Complete the following statement: A displacement vector a) is directed from an object’s final position toward its initial position. b) is always directed along a tangent to the object’s path. c) has a magnitude that always equals the distance the object traveled from its initial position to its final position. d) has SI units of meter per second. e) is directed from an object’s initial position toward its final position.

2.1.4. Complete the following statement: A displacement vector a) is directed from an object’s final position toward its initial position. b) is always directed along a tangent to the object’s path. c) has a magnitude that always equals the distance the object traveled from its initial position to its final position. d) has SI units of meter per second. e) is directed from an object’s initial position toward its final position.

2. 1. 1. In the morning, a bird is in Tampa, Florida 2.1.1. In the morning, a bird is in Tampa, Florida. In the afternoon, the bird is near Orlando, Florida. Given this information, which one of the following statements best describes the relationship between the magnitude of the bird’s displacement and the distance the bird traveled? a) The distance traveled is either greater than or equal to the magnitude of bird’s displacement. b) The distance traveled is either less than or equal to the magnitude of bird’s displacement. c) The distance traveled is equal to the magnitude of bird’s displacement. d) The distance traveled is either less than or greater than the magnitude of bird’s displacement. e) The distance traveled is greater than the magnitude of bird’s displacement.

2. 1. 1. In the morning, a bird is in Tampa, Florida 2.1.1. In the morning, a bird is in Tampa, Florida. In the afternoon, the bird is near Orlando, Florida. Given this information, which one of the following statements best describes the relationship between the magnitude of the bird’s displacement and the distance the bird traveled? a) The distance traveled is either greater than or equal to the magnitude of bird’s displacement. b) The distance traveled is either less than or equal to the magnitude of bird’s displacement. c) The distance traveled is equal to the magnitude of bird’s displacement. d) The distance traveled is either less than or greater than the magnitude of bird’s displacement. e) The distance traveled is greater than the magnitude of bird’s displacement.

2.1.2. A race car, traveling at constant speed, makes one lap around a circular track of radius r in a time t. The circumference of a circle is given by C = 2r. Which one of the following statements concerning this car is true? a) The displacement of the car does not change with time. b) The instantaneous velocity of the car is constant. c) The average speed of the car is the same over any time interval. d) The average velocity of the car is the same over any time interval. e) The average speed of the car over any time interval is equal to the magnitude of the average velocity over the same time interval.

2.1.2. A race car, traveling at constant speed, makes one lap around a circular track of radius r in a time t. The circumference of a circle is given by C = 2r. Which one of the following statements concerning this car is true? a) The displacement of the car does not change with time. b) The instantaneous velocity of the car is constant. c) The average speed of the car is the same over any time interval. d) The average velocity of the car is the same over any time interval. e) The average speed of the car over any time interval is equal to the magnitude of the average velocity over the same time interval.

2. 2. 1. A turtle and a rabbit are to have a race 2.2.1. A turtle and a rabbit are to have a race. The turtle’s average speed is 0.9 m/s. The rabbit’s average speed is 9 m/s. The distance from the starting line to the finish line is 1500 m. The rabbit decides to let the turtle run before he starts running to give the turtle a head start. What, approximately, is the maximum time the rabbit can wait before starting to run and still win the race? a) 15 minutes b) 18 minutes c) 20 minutes d) 22 minutes e) 25 minutes

SI units for speed: meters per second (m/s) 2.2 Speed and Velocity Average speed is the distance traveled divided by the time required to cover the distance. SI units for speed: meters per second (m/s)

Example 1 Distance Run by a Jogger 2.2 Speed and Velocity Example 1 Distance Run by a Jogger How far does a jogger run in 1.5 hours (5400 s) if his average speed is 2.22 m/s?

Average velocity is the displacement divided by the elapsed time. 2.2 Speed and Velocity Average velocity is the displacement divided by the elapsed time.

2.2 Speed and Velocity Example 2 The World’s Fastest Jet-Engine Car Andy Green in the car ThrustSSC set a world record of 341.1 m/s in 1997. To establish such a record, the driver makes two runs through the course, one in each direction, to nullify wind effects. From the data, determine the average velocity for each run.

2.2 Speed and Velocity

The instantaneous velocity indicates how fast 2.2 Speed and Velocity The instantaneous velocity indicates how fast the car moves and the direction of motion at each instant of time.

2.2.1. A motorcycle travels due south covering a total distance of 80.0 kilometers in 60.0 minutes. Which one of the following statements concerning this situation is necessarily true? a) The velocity of the motorcycle is constant. b) The acceleration of the motorcycle must be non-zero. c) The motorcycle traveled 40.0 kilometers during the first 30.0 minutes. d) The speed of the motorcycle must be 80.0 kilometers per hour throughout the entire trip. e) The average velocity of the motorcycle is 80.0 kilometers per hour, due south.

2.2.2. Which one of the following quantities is defined as the distance traveled divided by the elapsed time for the travel? a) average speed b) average velocity c) average acceleration d) instantaneous velocity e) instantaneous acceleration

2.2.2. Which one of the following quantities is defined as the distance traveled divided by the elapsed time for the travel? a) average speed b) average velocity c) average acceleration d) instantaneous velocity e) instantaneous acceleration

2.2.3. Which one of the following quantities is defined as an object’s displacement divided by the elapsed time for the displacement? a) average speed b) average velocity c) average acceleration d) instantaneous velocity e) instantaneous acceleration

2.2.3. Which one of the following quantities is defined as an object’s displacement divided by the elapsed time for the displacement? a) average speed b) average velocity c) average acceleration d) instantaneous velocity e) instantaneous acceleration

2. 2. 4. A train leaves a station, starting from rest, at time t = 0 2.2.4. A train leaves a station, starting from rest, at time t = 0.0 s. At time t = 3600.0 s, the train is traveling due west at 28 m/s. In this example, the 28 m/s is the train’s a) average speed b) average velocity c) average acceleration d) instantaneous velocity e) instantaneous acceleration

2. 2. 4. A train leaves a station, starting from rest, at time t = 0 2.2.4. A train leaves a station, starting from rest, at time t = 0.0 s. At time t = 3600.0 s, the train is traveling due west at 28 m/s. In this example, the 28 m/s is the train’s a) average speed b) average velocity c) average acceleration d) instantaneous velocity e) instantaneous acceleration

2.2.5. The speedometer on a car’s dashboard measures which of the following quantities? a) average speed b) average velocity c) average acceleration d) instantaneous velocity e) instantaneous acceleration

2.2.5. The speedometer on a car’s dashboard measures which of the following quantities? a) average speed b) average velocity c) average acceleration d) instantaneous velocity e) instantaneous acceleration

2. 2. 1. A turtle and a rabbit are to have a race 2.2.1. A turtle and a rabbit are to have a race. The turtle’s average speed is 0.9 m/s. The rabbit’s average speed is 9 m/s. The distance from the starting line to the finish line is 1500 m. The rabbit decides to let the turtle run before he starts running to give the turtle a head start. What, approximately, is the maximum time the rabbit can wait before starting to run and still win the race? a) 15 minutes b) 18 minutes c) 20 minutes d) 22 minutes e) 25 minutes

2. 2. 1. A turtle and a rabbit are to have a race 2.2.1. A turtle and a rabbit are to have a race. The turtle’s average speed is 0.9 m/s. The rabbit’s average speed is 9 m/s. The distance from the starting line to the finish line is 1500 m. The rabbit decides to let the turtle run before he starts running to give the turtle a head start. What, approximately, is the maximum time the rabbit can wait before starting to run and still win the race? a) 15 minutes b) 18 minutes c) 20 minutes d) 22 minutes e) 25 minutes

2. 2. 2. A turtle, moving at a constant velocity of 0 2.2.2. A turtle, moving at a constant velocity of 0.9 m/s due south, is in a race with a rabbit, who runs at a moderate speed of 9 m/s. When the turtle is 45 m from the finish line, the rabbit begins taunting the turtle by running from the turtle to the finish line (without crossing it) and back to the turtle. The rabbit continues going back and forth between the turtle and the finish line until the turtle crosses the finish line. About how many meters does the rabbit travel as the turtle travels that last 45 m? Assume the rabbit always runs at 9 m/s and doesn’t lose any time changing direction. a) 180 m b) 270 m c) 360 m d) 450 m e) 540 m

2. 2. 2. A turtle, moving at a constant velocity of 0 2.2.2. A turtle, moving at a constant velocity of 0.9 m/s due south, is in a race with a rabbit, who runs at a moderate speed of 9 m/s. When the turtle is 45 m from the finish line, the rabbit begins taunting the turtle by running from the turtle to the finish line (without crossing it) and back to the turtle. The rabbit continues going back and forth between the turtle and the finish line until the turtle crosses the finish line. About how many meters does the rabbit travel as the turtle travels that last 45 m? Assume the rabbit always runs at 9 m/s and doesn’t lose any time changing direction. a) 180 m b) 270 m c) 360 m d) 450 m e) 540 m

The notion of acceleration emerges when a change in velocity is combined with the time during which the change occurs.

DEFINITION OF AVERAGE ACCELERATION

2.3 Acceleration Example 3 Acceleration and Increasing Velocity Determine the average acceleration of the plane.

2.3 Acceleration

2.3 Acceleration Example 3 Acceleration and Decreasing Velocity

2.3 Acceleration

2.3.1. Which one of the following situations does the object have no acceleration? a) A ball at the end of a string is whirled in a horizontal circle at a constant speed. b) Seeing a red traffic light ahead, the driver of a minivan steps on the brake. As a result, the minivan slows from 15 m/s to stop before reaching the light. c) A boulder starts from rest and rolls down a mountain. d) An elevator in a tall skyscraper moves upward at a constant speed of 3 m/s. e) A ball is thrown upward from the surface of the earth, slows to a temporary stop at a height of 4 m, and begins to fall back toward the ground.

2.3.1. Which one of the following situations does the object have no acceleration? a) A ball at the end of a string is whirled in a horizontal circle at a constant speed. b) Seeing a red traffic light ahead, the driver of a minivan steps on the brake. As a result, the minivan slows from 15 m/s to stop before reaching the light. c) A boulder starts from rest and rolls down a mountain. d) An elevator in a tall skyscraper moves upward at a constant speed of 3 m/s. e) A ball is thrown upward from the surface of the earth, slows to a temporary stop at a height of 4 m, and begins to fall back toward the ground.

2.3.2. In which one of the following situations does the car have an acceleration that is directed due north? a) A car travels northward with a constant speed of 24 m/s. b) A car is traveling southward as its speed increases from 24 m/s to 33 m/s. c) A car is traveling southward as its speed decreases from 24 m/s to 18 m/s. d) A car is traveling northward as its speed decreases from 24 m/s to 18 m/s. e) A car travels southward with a constant speed of 24 m/s.

2.3.2. In which one of the following situations does the car have an acceleration that is directed due north? a) A car travels northward with a constant speed of 24 m/s. b) A car is traveling southward as its speed increases from 24 m/s to 33 m/s. c) A car is traveling southward as its speed decreases from 24 m/s to 18 m/s. d) A car is traveling northward as its speed decreases from 24 m/s to 18 m/s. e) A car travels southward with a constant speed of 24 m/s.

2.3.1. A ball is thrown toward a wall, bounces, and returns to the thrower with the same speed as it had before it bounced. Which one of the following statements correctly describes this situation? a) The ball was not accelerated during its contact with the wall because its speed remained constant. b) The instantaneous velocity of the ball from the time it left the thrower’s hand was constant. c) The only time that the ball had an acceleration was when the ball started from rest and left the hand of the thrower and again when the ball returned to the hand and was stopped. d) During this situation, the ball was never accelerated. e) The ball was accelerated during its contact with the wall because its direction changed.

2.3.1. A ball is thrown toward a wall, bounces, and returns to the thrower with the same speed as it had before it bounced. Which one of the following statements correctly describes this situation? a) The ball was not accelerated during its contact with the wall because its speed remained constant. b) The instantaneous velocity of the ball from the time it left the thrower’s hand was constant. c) The only time that the ball had an acceleration was when the ball started from rest and left the hand of the thrower and again when the ball returned to the hand and was stopped. d) During this situation, the ball was never accelerated. e) The ball was accelerated during its contact with the wall because its direction changed.

2. 3. 2. In an air race, two planes are traveling due east 2.3.2. In an air race, two planes are traveling due east. Plane One has a larger acceleration than Plane Two. Both accelerations are in the same direction. Which one of the following statements is true concerning this situation? a) In the same time interval, the change in the velocity of the Plane Two is greater than that of Plane One. b) In the same time interval, the change in the velocity of the Plane One is greater than that of Plane Two. c) Within the time interval, the velocity of the Plane Two remains greater than that of Plane One. d) Within the time interval, the velocity of the Plane One remains greater than that of Plane Two. e) Too little information is given to compare the velocities of the planes or how the velocities are changing.

2. 3. 2. In an air race, two planes are traveling due east 2.3.2. In an air race, two planes are traveling due east. Plane One has a larger acceleration than Plane Two. Both accelerations are in the same direction. Which one of the following statements is true concerning this situation? a) In the same time interval, the change in the velocity of the Plane Two is greater than that of Plane One. b) In the same time interval, the change in the velocity of the Plane One is greater than that of Plane Two. c) Within the time interval, the velocity of the Plane Two remains greater than that of Plane One. d) Within the time interval, the velocity of the Plane One remains greater than that of Plane Two. e) Too little information is given to compare the velocities of the planes or how the velocities are changing.

2. 3. 3. Two cars travel along a level highway 2.3.3. Two cars travel along a level highway. An observer notices that the distance between the cars is increasing. Which one of the following statements concerning this situation is necessarily true? a) Both cars could be accelerating at the same rate. b) The leading car has the greater acceleration. c) The trailing car has the smaller acceleration. d) The velocity of each car is increasing. e) At least one of the cars has a non-zero acceleration.

2. 3. 3. Two cars travel along a level highway 2.3.3. Two cars travel along a level highway. An observer notices that the distance between the cars is increasing. Which one of the following statements concerning this situation is necessarily true? a) Both cars could be accelerating at the same rate. b) The leading car has the greater acceleration. c) The trailing car has the smaller acceleration. d) The velocity of each car is increasing. e) At least one of the cars has a non-zero acceleration.

2.3.4. A police cruiser is parked by the side of the road when a speeding car passes. The cruiser follows the speeding car. Consider the following diagrams where the dots represent the cruiser’s position at 0.5-s intervals. Which diagram(s) are possible representations of the cruiser’s motion? a) A only b) B, D, or E only c) C only d) E only e) A or C only

2.3.4. A police cruiser is parked by the side of the road when a speeding car passes. The cruiser follows the speeding car. Consider the following diagrams where the dots represent the cruiser’s position at 0.5-s intervals. Which diagram(s) are possible representations of the cruiser’s motion? a) A only b) B, D, or E only c) C only d) E only e) A or C only

2.4 Equations of Kinematics for Constant Acceleration It is customary to dispense with the use of boldface symbols overdrawn with arrows for the displacement, velocity, and acceleration vectors. We will, however, continue to convey the directions with a plus or minus sign.

2.4 Equations of Kinematics for Constant Acceleration Let the object be at the origin when the clock starts.

2.4 Equations of Kinematics for Constant Acceleration

2.4 Equations of Kinematics for Constant Acceleration Five kinematic variables: 1. displacement, x 2. acceleration (constant), a 3. final velocity (at time t), v 4. initial velocity, vo 5. elapsed time, t

2.4 Equations of Kinematics for Constant Acceleration

2.4 Equations of Kinematics for Constant Acceleration

2.4 Equations of Kinematics for Constant Acceleration Example 6 Catapulting a Jet Find its displacement.

2.4 Equations of Kinematics for Constant Acceleration

2.4 Equations of Kinematics for Constant Acceleration

2.4 Equations of Kinematics for Constant Acceleration

2.5 Applications of the Equations of Kinematics Reasoning Strategy 1. Make a drawing. 2. Decide which directions are to be called positive (+) and negative (-). 3. Write down the values that are given for any of the five kinematic variables. 4. Verify that the information contains values for at least three of the five kinematic variables. Select the appropriate equation. 5. When the motion is divided into segments, remember that the final velocity of one segment is the initial velocity for the next. 6. Keep in mind that there may be two possible answers to a kinematics problem.

2.5 Applications of the Equations of Kinematics Example 8 An Accelerating Spacecraft A spacecraft is traveling with a velocity of +3250 m/s. Suddenly the retrorockets are fired, and the spacecraft begins to slow down with an acceleration whose magnitude is 10.0 m/s2. What is the velocity of the spacecraft when the displacement of the craft is +215 km, relative to the point where the retrorockets began firing? x a v vo t +215000 m -10.0 m/s2 ? +3250 m/s

2.5 Applications of the Equations of Kinematics

2.5 Applications of the Equations of Kinematics x a v vo t +215000 m -10.0 m/s2 ? +3250 m/s

In the absence of air resistance, it is found that all bodies 2.6 Freely Falling Bodies In the absence of air resistance, it is found that all bodies at the same location above the Earth fall vertically with the same acceleration. If the distance of the fall is small compared to the radius of the Earth, then the acceleration remains essentially constant throughout the descent. This idealized motion is called free-fall and the acceleration of a freely falling body is called the acceleration due to gravity.

2.6 Freely Falling Bodies

Example 10 A Falling Stone 2.6 Freely Falling Bodies Example 10 A Falling Stone A stone is dropped from the top of a tall building. After 3.00s of free fall, what is the displacement y of the stone?

2.6 Freely Falling Bodies y a v vo t ? -9.80 m/s2 0 m/s 3.00 s

2.6 Freely Falling Bodies y a v vo t ? -9.80 m/s2 0 m/s 3.00 s

Example 12 How High Does it Go? The referee tosses the coin up 2.6 Freely Falling Bodies Example 12 How High Does it Go? The referee tosses the coin up with an initial speed of 5.00m/s. In the absence if air resistance, how high does the coin go above its point of release?

2.6 Freely Falling Bodies y a v vo t ? -9.80 m/s2 0 m/s +5.00 m/s

2.6 Freely Falling Bodies y a v vo t ? -9.80 m/s2 0 m/s +5.00 m/s

2.6.1. Two identical ping-pong balls are selected for a physics demonstration. A tiny hole is drilled in one of the balls; and the ball is filled with water. The hole is sealed so that no water can escape. The two balls are then dropped from rest at the exact same time from the roof of a building. Assuming there is no wind, which one of the following statements is true? a) The two balls reach the ground at the same time. b) The heavier ball reaches the ground a long time before the lighter ball. c) The heavier ball reaches the ground just before the lighter ball. d) The heavier ball has a much larger velocity when it strikes the ground than the light ball. e) The heavier ball has a slightly larger velocity when it strikes the ground than the light ball.

2.6.1. Two identical ping-pong balls are selected for a physics demonstration. A tiny hole is drilled in one of the balls; and the ball is filled with water. The hole is sealed so that no water can escape. The two balls are then dropped from rest at the exact same time from the roof of a building. Assuming there is no wind, which one of the following statements is true? a) The two balls reach the ground at the same time. b) The heavier ball reaches the ground a long time before the lighter ball. c) The heavier ball reaches the ground just before the lighter ball. d) The heavier ball has a much larger velocity when it strikes the ground than the light ball. e) The heavier ball has a slightly larger velocity when it strikes the ground than the light ball.

2.6.2. Two identical ping-pong balls are selected for a physics demonstration. A tiny hole is drilled in one of the balls; and the ball is filled with water. The hole is sealed so that no water can escape. Each ball is shot horizontally from a gun with an initial velocity v0 from the top of a building. The following drawing shows several trajectories numbered 1 through 5. Which of the following statements is true? a) Both balls would follow trajectory 5. b) Both balls would follow trajectory 3. c) The lighter ball would follow 4 and the heavier ball would follow 2. d) The lighter ball would follow 4 and the heavier ball would follow 3. e) The lighter ball would follow 4 or 3 and the heavier ball would follow 2 or 1, depending on the magnitude of v0.

2.6.2. Two identical ping-pong balls are selected for a physics demonstration. A tiny hole is drilled in one of the balls; and the ball is filled with water. The hole is sealed so that no water can escape. Each ball is shot horizontally from a gun with an initial velocity v0 from the top of a building. The following drawing shows several trajectories numbered 1 through 5. Which of the following statements is true? a) Both balls would follow trajectory 5. b) Both balls would follow trajectory 3. c) The lighter ball would follow 4 and the heavier ball would follow 2. d) The lighter ball would follow 4 and the heavier ball would follow 3. e) The lighter ball would follow 4 or 3 and the heavier ball would follow 2 or 1, depending on the magnitude of v0.

2.6.3. A cannon directed straight upward launches a ball with an initial speed v. The ball reaches a maximum height h in a time t. Then, the same cannon is used to launch a second ball straight upward at a speed 2v. In terms of h and t, what is the maximum height the second ball reaches and how long does it take to reach that height? a) 2h, t b) 4h, 2t c) 2h, 4t d) 2h, 2t e) h, t

2.6.3. A cannon directed straight upward launches a ball with an initial speed v. The ball reaches a maximum height h in a time t. Then, the same cannon is used to launch a second ball straight upward at a speed 2v. In terms of h and t, what is the maximum height the second ball reaches and how long does it take to reach that height? a) 2h, t b) 4h, 2t c) 2h, 4t d) 2h, 2t e) h, t

Conceptual Example 14 Acceleration Versus Velocity 2.6 Freely Falling Bodies Conceptual Example 14 Acceleration Versus Velocity There are three parts to the motion of the coin. On the way up, the coin has a vector velocity that is directed upward and has decreasing magnitude. At the top of its path, the coin momentarily has zero velocity. On the way down, the coin has downward-pointing velocity with an increasing magnitude. In the absence of air resistance, does the acceleration of the coin, like the velocity, change from one part to another?

Conceptual Example 15 Taking Advantage of Symmetry 2.6 Freely Falling Bodies Conceptual Example 15 Taking Advantage of Symmetry Does the pellet in part b strike the ground beneath the cliff with a smaller, greater, or the same speed as the pellet in part a?

2.6.1. A rock is released from rest from a hot air balloon that is at rest with respect to the ground a few meters below. If we ignore air resistance as the rock falls, which one of the following statements is true? a) The rock will take longer than one second to reach the ground. b) The instantaneous speed of the rock just before it reaches the ground will be 9.8 m/s. c) The rock is considered a freely falling body after it is released. d) As the rock falls, its acceleration is 9.8 m/s2, directed upward. e) After the ball is released it falls at a constant speed of 9.8 m/s.

2.6.1. A rock is released from rest from a hot air balloon that is at rest with respect to the ground a few meters below. If we ignore air resistance as the rock falls, which one of the following statements is true? a) The rock will take longer than one second to reach the ground. b) The instantaneous speed of the rock just before it reaches the ground will be 9.8 m/s. c) The rock is considered a freely falling body after it is released. d) As the rock falls, its acceleration is 9.8 m/s2, directed upward. e) After the ball is released it falls at a constant speed of 9.8 m/s.

2. 6. 2. Ping-pong ball A is filled with sand 2.6.2. Ping-pong ball A is filled with sand. Ping-pong ball B is identical to A, except that it is empty inside. Ball A is somewhat heavier than ball B because of the sand inside. Both balls are simultaneously dropped from rest from the top of a building. Which of these two balls has the greater acceleration due to gravity, if any, as they fall? a) ball A b) ball B c) Both ball A and ball B have zero acceleration. d) Both ball A and ball B have the same acceleration.

2. 6. 2. Ping-pong ball A is filled with sand 2.6.2. Ping-pong ball A is filled with sand. Ping-pong ball B is identical to A, except that it is empty inside. Ball A is somewhat heavier than ball B because of the sand inside. Both balls are simultaneously dropped from rest from the top of a building. Which of these two balls has the greater acceleration due to gravity, if any, as they fall? a) ball A b) ball B c) Both ball A and ball B have zero acceleration. d) Both ball A and ball B have the same acceleration.

2.6.3. A ball is thrown vertically upward from the surface of the earth. The ball rises to some maximum height and falls back toward the surface of the earth. Which one of the following statements concerning this situation is true if air resistance is neglected? a) As the ball rises, its acceleration vector points upward. b) The ball is a freely falling body for the duration of its flight. c) The acceleration of the ball is zero when the ball is at its highest point. d) The speed of the ball is negative while the ball falls back toward the earth. e) The velocity and acceleration of the ball always point in the same direction.

2.6.3. A ball is thrown vertically upward from the surface of the earth. The ball rises to some maximum height and falls back toward the surface of the earth. Which one of the following statements concerning this situation is true if air resistance is neglected? a) As the ball rises, its acceleration vector points upward. b) The ball is a freely falling body for the duration of its flight. c) The acceleration of the ball is zero when the ball is at its highest point. d) The speed of the ball is negative while the ball falls back toward the earth. e) The velocity and acceleration of the ball always point in the same direction.

2.7 Graphical Analysis of Velocity and Acceleration

2.7 Graphical Analysis of Velocity and Acceleration

2.7 Graphical Analysis of Velocity and Acceleration

2.7 Graphical Analysis of Velocity and Acceleration

2. 7. 1. A dog is initially walking due east 2.7.1. A dog is initially walking due east. He stops, noticing a cat behind him. He runs due west and stops when the cat disappears into some bushes. He starts walking due east again. Then, a motorcycle passes him and he runs due east after it. The dog gets tired and stops running. Which of the following graphs correctly represent the position versus time of the dog?

2. 7. 1. A dog is initially walking due east 2.7.1. A dog is initially walking due east. He stops, noticing a cat behind him. He runs due west and stops when the cat disappears into some bushes. He starts walking due east again. Then, a motorcycle passes him and he runs due east after it. The dog gets tired and stops running. Which of the following graphs correctly represent the position versus time of the dog?

2.7.2. The graph below represents the speed of a car traveling due east for a portion of its travel along a horizontal road. Which of the following statements concerning this graph is true? a) The car initially increases its speed, but then the speed decreases at a constant rate until the car stops. b) The speed of the car is initially constant, but then it has a variable positive acceleration before it stops. c) The car initially has a positive acceleration, but then it has a variable negative acceleration before it stops. d) The car initially has a positive acceleration, but then it has a variable positive acceleration before it stops. e) No information about the acceleration of the car can be determined from this graph.

2.7.2. The graph below represents the speed of a car traveling due east for a portion of its travel along a horizontal road. Which of the following statements concerning this graph is true? a) The car initially increases its speed, but then the speed decreases at a constant rate until the car stops. b) The speed of the car is initially constant, but then it has a variable positive acceleration before it stops. c) The car initially has a positive acceleration, but then it has a variable negative acceleration before it stops. d) The car initially has a positive acceleration, but then it has a variable positive acceleration before it stops. e) No information about the acceleration of the car can be determined from this graph.

2. 7. 1. A dog is walking along a street 2.7.1. A dog is walking along a street. As the dog moves, a graph is made of its position on the vertical axis with the elapsed time on the horizontal axis. The slope of the curve is determined at some point on the graph. The slope of this curve is a measurement of which of the following parameters? a) the dog’s instantaneous velocity b) the dog’s acceleration c) the dog’s speed d) the dog’s average velocity e) the elapsed time for the dog’s walk

2. 7. 1. A dog is walking along a street 2.7.1. A dog is walking along a street. As the dog moves, a graph is made of its position on the vertical axis with the elapsed time on the horizontal axis. The slope of the curve is determined at some point on the graph. The slope of this curve is a measurement of which of the following parameters? a) the dog’s instantaneous velocity b) the dog’s acceleration c) the dog’s speed d) the dog’s average velocity e) the elapsed time for the dog’s walk

2.7.2. Starting from rest, a particle that is confined to move along a straight line is accelerated at a rate of 5.0 m/s2. Which one of the following statements concerning the slope of the position versus time graph for this particle is true? a) The slope has a constant value of 5.0 m/s. b) The slope has a constant value of 5.0 m/s2. c) The slope is both constant and negative. d) The slope is not constant and increases with increasing time. e) The slope is not constant and decreases with increasing time.

2.7.2. Starting from rest, a particle that is confined to move along a straight line is accelerated at a rate of 5.0 m/s2. Which one of the following statements concerning the slope of the position versus time graph for this particle is true? a) The slope has a constant value of 5.0 m/s. b) The slope has a constant value of 5.0 m/s2. c) The slope is both constant and negative. d) The slope is not constant and increases with increasing time. e) The slope is not constant and decreases with increasing time.

2.7.3. The graph shows the velocity of an object versus the elapsed time. During which interval on the graph does the object’s acceleration decrease with time? a) A b) B c) C d) D e) E

2.7.3. The graph shows the velocity of an object versus the elapsed time. During which interval on the graph does the object’s acceleration decrease with time? a) A b) B c) C d) D e) E

2.7.4. Complete the following statement: the instantaneous acceleration of an object can be determined by determining the slope of a) the object’s velocity versus elapsed time graph. b) the object’s displacement versus elapsed time graph. c) the object’s distance versus elapsed time graph. d) the object’s acceleration versus elapsed time graph.

2.7.4. Complete the following statement: the instantaneous acceleration of an object can be determined by determining the slope of a) the object’s velocity versus elapsed time graph. b) the object’s displacement versus elapsed time graph. c) the object’s distance versus elapsed time graph. d) the object’s acceleration versus elapsed time graph.