Kinematics in Two Dimensions Chapter 3. 3.1 Displacement, Velocity, and Acceleration.

Kinematics in Two Dimensions

Chapter 3

3.1 Displacement, Velocity, and Acceleration

position initial or

position final r

ntdisplaceme orrr


ttt o

o

rrr

v

Average velocity is the displacement divided by the elapsed time.


The instantaneous velocity indicates how fastthe car moves and the direction of motion at eachinstant of time.

tt

rv

0lim


tt

rv

0lim


ttt o

o

vvv

a

DEFINITION OF AVERAGE ACCELERATION

ov

v v

3.1.1. Which one of the following statements concerning the displacement of an object is false?

a) Displacement is a vector quantity that points from the initial position of an object to its final position.

b) The magnitude of an object’s displacement is always equal to the distance it traveled from its initial position to its final position.

c) The magnitude of an object’s displacement is the shortest distance from its initial position to its final position.

d) The direction of an object’s displacement is indicated by an arrow that begins on the initial position of the object and ends on its final position.

e) The length of the arrow representing an object’s displacement is proportional to its magnitude.

3.1.1. Which one of the following statements concerning the displacement of an object is false?

a) Displacement is a vector quantity that points from the initial position of an object to its final position.

b) The magnitude of an object’s displacement is always equal to the distance it traveled from its initial position to its final position.

c) The magnitude of an object’s displacement is the shortest distance from its initial position to its final position.

d) The direction of an object’s displacement is indicated by an arrow that begins on the initial position of the object and ends on its final position.

e) The length of the arrow representing an object’s displacement is proportional to its magnitude.

3.1.2. At time t = 0 s, the position vector of a sailboat is r0. Later, at time t, the sailboat has a position vector r. Which of the following expressions correctly indicates the displacement of the sailboat during the time interval, t t0?

a) r

b) r0

c) r + r0

d) r r0

e) r0 r

3.1.2. At time t = 0 s, the position vector of a sailboat is r0. Later, at time t, the sailboat has a position vector r. Which of the following expressions correctly indicates the displacement of the sailboat during the time interval, t t0?

a) r

b) r0

c) r + r0

d) r r0

e) r0 r

3.1.3. A park ranger wanted to measure the height of a tall tree. The ranger stood 6.10 m from the base of the tree; and he observed that his line of sight made an angle of 73.5° above the horizontal as he looked at the top of the tree. What is the height of the tree?

a) 5.84 m

b) 8.77 m

c) 11.7 m

d) 17.3 m

e) 20.6 m

3.1.3. A park ranger wanted to measure the height of a tall tree. The ranger stood 6.10 m from the base of the tree; and he observed that his line of sight made an angle of 73.5° above the horizontal as he looked at the top of the tree. What is the height of the tree?

a) 5.84 m

b) 8.77 m

c) 11.7 m

d) 17.3 m

e) 20.6 m

3.1.4. Which one of the following quantities is an object’s displacement divided by the elapsed time of the displacement?

a) average velocity

b) instantaneous velocity

c) average displacement

d) average acceleration

e) instantaneous acceleration

3.1.4. Which one of the following quantities is an object’s displacement divided by the elapsed time of the displacement?

a) average velocity





3.1.5. Which one of the following quantities is the change in object’s velocity divided by the elapsed time as the elapsed time becomes very small?

a) average velocity





3.1.5. Which one of the following quantities is the change in object’s velocity divided by the elapsed time as the elapsed time becomes very small?

a) average velocity





3.1.6. How is the direction of the average acceleration determined?

a) The direction of the average acceleration is the same as that of the displacement vector.

b) The direction of the average acceleration is the same as that of the instantaneous velocity vector.

c) The direction of the average acceleration is that of the vector subtraction of the initial velocity from the final velocity.

d) The direction of the average acceleration is the same as that of the average velocity vector.

e) The direction of the average acceleration is that of the vector addition of the initial velocity from the final velocity.

3.1.6. How is the direction of the average acceleration determined?

a) The direction of the average acceleration is the same as that of the displacement vector.

b) The direction of the average acceleration is the same as that of the instantaneous velocity vector.

c) The direction of the average acceleration is that of the vector subtraction of the initial velocity from the final velocity.

d) The direction of the average acceleration is the same as that of the average velocity vector.

e) The direction of the average acceleration is that of the vector addition of the initial velocity from the final velocity.

3.1.7. A delivery truck leaves a warehouse and travels 3.20 km east. The truck makes a right turn and travels 2.45 km south to arrive at its destination. What is the magnitude and direction of the truck’s displacement from the warehouse?

a) 4.03 km, 37.4 south of east

b) 2.30 km, 52.5 south of east

c) 0.75 km, 37.8 south of east

d) 2.40 km, 45.0 south of east

e) 5.65 km, 52.5 south of east

3.1.7. A delivery truck leaves a warehouse and travels 3.20 km east. The truck makes a right turn and travels 2.45 km south to arrive at its destination. What is the magnitude and direction of the truck’s displacement from the warehouse?

a) 4.03 km, 37.4 south of east

b) 2.30 km, 52.5 south of east

c) 0.75 km, 37.8 south of east

d) 2.40 km, 45.0 south of east

e) 5.65 km, 52.5 south of east

3.1.8. While on a one-hour trip, a small boat travels 32 km north and then travels 45 km east. What is the boat's average speed for the one-hour trip?

a) 39 km/h

b) 55 km/h

c) 77 km/h

d) 89 km/h

e) 96 km/h

3.1.8. While on a one-hour trip, a small boat travels 32 km north and then travels 45 km east. What is the boat's average speed for the one-hour trip?

a) 39 km/h

b) 55 km/h

c) 77 km/h

d) 89 km/h

e) 96 km/h

3.1.9. While on a one-hour trip, a small boat travels 33 km north and then travels 45 km east. What is the direction of the boat's average velocity for the one-hour trip?

a) 45 north of east

b) 54 north of east

c) 37 north of east

d) 27 north of east

e) due east

3.1.9. While on a one-hour trip, a small boat travels 33 km north and then travels 45 km east. What is the direction of the boat's average velocity for the one-hour trip?

a) 45 north of east

b) 54 north of east

c) 37 north of east

d) 27 north of east

e) due east

3.1.1. A truck drives due south for 1.2 km in 1.5 minutes. Then, the truck turns and drives due west for 1.2 km in 1.5 minutes. Which one of the following statements is correct?

a) The average speed for the two segments is the same. The average velocity for the two segments is the same.

b) The average speed for the two segments is not the same. The average velocity for the two segments is the same.

c) The average speed for the two segments is the same. The average velocity for the two segments is not the same.

d) The average speed for the two segments is not the same. The average velocity for the two segments is not the same.

3.1.1. A truck drives due south for 1.2 km in 1.5 minutes. Then, the truck turns and drives due west for 1.2 km in 1.5 minutes. Which one of the following statements is correct?

a) The average speed for the two segments is the same. The average velocity for the two segments is the same.

b) The average speed for the two segments is not the same. The average velocity for the two segments is the same.

c) The average speed for the two segments is the same. The average velocity for the two segments is not the same.

d) The average speed for the two segments is not the same. The average velocity for the two segments is not the same.

3.1.2. A ball is rolling down one hill and up another as shown. Points A and B are at the same height. How do the velocity and acceleration change as the ball rolls from point A to point B?

a) The velocity and acceleration are the same at both points.

b) The velocity and the magnitude of the acceleration are the same at both points, but the direction of the acceleration is opposite at B to the direction it had at A.

c) The acceleration and the magnitude of the velocity are the same at both points, but the direction of the velocity is opposite at B to the direction it had at A.

d) The horizontal component of the velocity is the same at points A and B, but the vertical component of the velocity has the same magnitude, but the opposite sign at B. The acceleration at points A and B is the same.

e) The vertical component of the velocity is the same at points A and B, but the horizontal component of the velocity has the same magnitude, but the opposite sign at B. The acceleration at points A and B has the same magnitude, but opposite direction.

3.1.2. A ball is rolling down one hill and up another as shown. Points A and B are at the same height. How do the velocity and acceleration change as the ball rolls from point A to point B?

a) The velocity and acceleration are the same at both points.

b) The velocity and the magnitude of the acceleration are the same at both points, but the direction of the acceleration is opposite at B to the direction it had at A.

c) The acceleration and the magnitude of the velocity are the same at both points, but the direction of the velocity is opposite at B to the direction it had at A.

d) The horizontal component of the velocity is the same at points A and B, but the vertical component of the velocity has the same magnitude, but the opposite sign at B. The acceleration at points A and B is the same.

e) The vertical component of the velocity is the same at points A and B, but the horizontal component of the velocity has the same magnitude, but the opposite sign at B. The acceleration at points A and B has the same magnitude, but opposite direction.

3.2 Equations of Kinematics in Two Dimensions

Equations of Kinematics

tvvx o 21

221 attvx o

atvv o

axvv o 222


tavv xoxx tvvx xox 21

xavv xoxx 222 221 tatvx xox


tavv yoyy

221 tatvy yoy

tvvy yoy 21

yavv yoyy 222


The x part of the motion occurs exactly as it would if the y part did not occur at all, and vice versa.


Example 1 A Moving Spacecraft

In the x direction, the spacecraft has an initial velocity componentof +22 m/s and an acceleration of +24 m/s2. In the y direction, theanalogous quantities are +14 m/s and an acceleration of +12 m/s2.Find (a) x and vx, (b) y and vy, and (c) the final velocity of thespacecraft at time 7.0 s.


Reasoning Strategy1. 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 fivekinematic variables associated with each direction.

4. Verify that the information contains values for at least threeof the kinematic variables. Do this for x and y. Select the appropriate equation.

5. When the motion is divided into segments, remember thatthe 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.


Example 1 A Moving Spacecraft

In the x direction, the spacecraft has an initial velocity componentof +22 m/s and an acceleration of +24 m/s2. In the y direction, theanalogous quantities are +14 m/s and an acceleration of +12 m/s2.Find (a) x and vx, (b) y and vy, and (c) the final velocity of thespacecraft at time 7.0 s.

x ax vx vox t? +24.0 m/s2 ? +22 m/s 7.0 s

y ay vy voy t? +12.0 m/s2 ? +14 m/s 7.0 s


x ax vx vox t? +24.0 m/s2 ? +22 m/s 7.0 s

m 740s 0.7sm24s 0.7sm22 2221

221

tatvx xox

sm190s 0.7sm24sm22 2

tavv xoxx


y ay vy voy t? +12.0 m/s2 ? +14 m/s 7.0 s

m 390s 0.7sm12s 0.7sm14 2221

221

tatvy yoy

sm98s 0.7sm12sm14 2

tavv yoyy


vsm98yv

sm190xv

sm210sm98sm190 22 v

2719098tan 1


3.2.1. In two-dimensional motion in the x-y plane, what is the relationship between the x part of the motion to the y part of the motion?

a) The x part of the motion is independent of the y part of the motion.

b) The y part of the motion goes as the square of the x part of the motion.

c) The x part of the motion is linearly dependent on the y part of the motion.

d) The x part of the motion goes as the square of the y part of the motion.

e) If the y part of the motion is in the vertical direction, then x part of the motion is dependent on the y part.

3.2.1. In two-dimensional motion in the x-y plane, what is the relationship between the x part of the motion to the y part of the motion?

a) The x part of the motion is independent of the y part of the motion.

b) The y part of the motion goes as the square of the x part of the motion.

c) The x part of the motion is linearly dependent on the y part of the motion.

d) The x part of the motion goes as the square of the y part of the motion.

e) If the y part of the motion is in the vertical direction, then x part of the motion is dependent on the y part.

3.2.2. Complete the following statement: In two-dimensional motion in the x-y plane, the x part of the motion and the y part of the motion are independent

a) only if there is no acceleration in either direction.

b) only if there is no acceleration in one of the directions.

c) only if there is an acceleration in both directions.

d) whether or not there is an acceleration in any direction.

e) whenever the acceleration is in the y direction only.

3.2.2. Complete the following statement: In two-dimensional motion in the x-y plane, the x part of the motion and the y part of the motion are independent

a) only if there is no acceleration in either direction.

b) only if there is no acceleration in one of the directions.

c) only if there is an acceleration in both directions.

d) whether or not there is an acceleration in any direction.

e) whenever the acceleration is in the y direction only.

3.2.1. An eagle takes off from a tree branch on the side of a mountain and flies due west for 225 m in 19 s. Spying a mouse on the ground to the west, the eagle dives 441 m at an angle of 65 relative to the horizontal direction for 11 s to catch the mouse. Determine the eagle’s average velocity for the thirty second interval.

a) 19 m/s at 44 below the horizontal direction

b) 22 m/s at 65 below the horizontal direction

c) 19 m/s at 65 below the horizontal direction

d) 22 m/s at 44 below the horizontal direction

e) 25 m/s at 27 below the horizontal direction

3.2.1. An eagle takes off from a tree branch on the side of a mountain and flies due west for 225 m in 19 s. Spying a mouse on the ground to the west, the eagle dives 441 m at an angle of 65 relative to the horizontal direction for 11 s to catch the mouse. Determine the eagle’s average velocity for the thirty second interval.

a) 19 m/s at 44 below the horizontal direction

b) 22 m/s at 65 below the horizontal direction

c) 19 m/s at 65 below the horizontal direction

d) 22 m/s at 44 below the horizontal direction

e) 25 m/s at 27 below the horizontal direction

3.2.2. A space craft is initially traveling toward Mars. As the craft approaches the planet, rockets are fired and the spacecraft temporarily stops and reorients itself. Then, at time t = 0 s, the rockets again fire causing the craft to move toward Mars with a constant acceleration. At time t, the craft’s displacement is r and its velocity v. Assuming the acceleration is constant, what would be its displacement and velocity at time 3t?

a) 3r and 3v

b) 4r and 2v

c) 6r and 3v

d) 9r and 3v

e) 9r and 6v

3.2.2. A space craft is initially traveling toward Mars. As the craft approaches the planet, rockets are fired and the spacecraft temporarily stops and reorients itself. Then, at time t = 0 s, the rockets again fire causing the craft to move toward Mars with a constant acceleration. At time t, the craft’s displacement is r and its velocity v. Assuming the acceleration is constant, what would be its displacement and velocity at time 3t?

a) 3r and 3v

b) 4r and 2v

c) 6r and 3v

d) 9r and 3v

e) 9r and 6v

3.2.3. Cathy and Jim have an argument about which route is the fastest route between their home at point A in the drawing and their workplace at point B. Cathy drives east and then north to work with a stop sign at the turn. Jim goes north, stops at a stop sign, and then goes northeast before reaching another stop sign, at which he makes a right turn to go east. Their cars are identical; each accelerates from rest to the maximum speed on either route of 15.6 m/s in 7.74 s. For each segment, they accelerate to the maximum speed, drive at that speed, and then decelerate at a rate of 2.5 m/s2 before each stop. Who gets to work first and what is his/her average velocity? The distances of the sides labeled “a” are 1.00 km and those labeled “b” are 6.00 km.

a) They arrive at the same time with an average velocity of 12.5 m/s, 45 north of east.

b) Jim arrives first with an average velocity of 14.1 m/s, 45 north of east.

c) Cathy arrives first with an average velocity of 12.5 m/s, 45 north of east.

d) Jim arrives first with an average velocity of 11.4 m/s, 45 north of east.

e) Cathy arrives first with an average velocity of 10.8 m/s, 45 north of east.

3.2.3. Cathy and Jim have an argument about which route is the fastest route between their home at point A in the drawing and their workplace at point B. Cathy drives east and then north to work with a stop sign at the turn. Jim goes north, stops at a stop sign, and then goes northeast before reaching another stop sign, at which he makes a right turn to go east. Their cars are identical; each accelerates from rest to the maximum speed on either route of 15.6 m/s in 7.74 s. For each segment, they accelerate to the maximum speed, drive at that speed, and then decelerate at a rate of 2.5 m/s2 before each stop. Who gets to work first and what is his/her average velocity? The distances of the sides labeled “a” are 1.00 km and those labeled “b” are 6.00 km.

a) They arrive at the same time with an average velocity of 12.5 m/s, 45 north of east.

b) Jim arrives first with an average velocity of 14.1 m/s, 45 north of east.

c) Cathy arrives first with an average velocity of 12.5 m/s, 45 north of east.

d) Jim arrives first with an average velocity of 11.4 m/s, 45 north of east.

e) Cathy arrives first with an average velocity of 10.8 m/s, 45 north of east.

3.3 Projectile Motion

Under the influence of gravity alone, an object near the surface of the Earth will accelerate downwards at 9.80m/s2.

2sm80.9ya 0xa

constant oxx vv


Example 3 A Falling Care Package

The airplane is moving horizontally with a constant velocity of +115 m/s at an altitude of 1050m. Determine the time requiredfor the care package to hit the ground.


y ay vy voy t-1050 m -9.80 m/s2 0 m/s ?


y ay vy voy t-1050 m -9.80 m/s2 0 m/s ?

221 tatvy yoy 2

21 tay y

s 6.14

sm9.80

m 1050222

ya

yt


Example 4 The Velocity of the Care Package

What are the magnitude and direction of the final velocity ofthe care package?


y ay vy voy t-1050 m -9.80 m/s2 ? 0 m/s 14.6 s


y ay vy voy t-1050 m -9.80 m/s2 ? 0 m/s 14.6 s

sm143

s 6.14sm80.90 2

tavv yoyy


Conceptual Example 5 I Shot a Bullet into the Air...

Suppose you are driving a convertible with the top down.The car is moving to the right at constant velocity. You pointa rifle straight up into the air and fire it. In the absence of airresistance, where would the bullet land – behind you, aheadof you, or in the barrel of the rifle?


Example 6 The Height of a Kickoff

A placekicker kicks a football at and angle of 40.0 degrees andthe initial speed of the ball is 22 m/s. Ignoring air resistance, determine the maximum height that the ball attains.


ov

oxv

oyv

sm1440sinsm22sin ooy vv

sm1740cossm22sin oox vv


y ay vy voy t? -9.80 m/s2 0 14 m/s


y ay vy voy t? -9.80 m/s2 0 14 m/s

yavv yoyy 222 y

oyy

a

vvy

2

22

m 10

sm8.92

sm1402

2

y


Example 7 The Time of Flight of a Kickoff

What is the time of flight between kickoff and landing?


y ay vy voy t0 -9.80 m/s2 14 m/s ?


y ay vy voy t0 -9.80 m/s2 14 m/s ?

221 tatvy yoy

2221 sm80.9sm140 tt

t2sm80.9sm1420

s 9.2t


Example 8 The Range of a Kickoff

Calculate the range R of the projectile.

m 49s 9.2sm17

221

tvtatvx oxxox


Conceptual Example 10 Two Ways to Throw a Stone

From the top of a cliff, a person throws two stones. The stoneshave identical initial speeds, but stone 1 is thrown downwardat some angle above the horizontal and stone 2 is thrown atthe same angle below the horizontal. Neglecting air resistance,which stone, if either, strikes the water with greater velocity?

3.3.1. A football is kicked at an angle 25 with respect to the horizontal. Which one of the following statements best describes the acceleration of the football during this event if air resistance is neglected?

a) The acceleration is zero m/s2 at all times.

b) The acceleration is zero m/s2 when the football has reached the highest point in its trajectory.

c) The acceleration is positive as the football rises, and it is negative as the football falls.

d) The acceleration starts at 9.8 m/s2 and drops to some constant lower value as the ball approaches the ground.

e) The acceleration is 9.8 m/s2 at all times.

3.3.1. A football is kicked at an angle 25 with respect to the horizontal. Which one of the following statements best describes the acceleration of the football during this event if air resistance is neglected?

a) The acceleration is zero m/s2 at all times.

b) The acceleration is zero m/s2 when the football has reached the highest point in its trajectory.

c) The acceleration is positive as the football rises, and it is negative as the football falls.

d) The acceleration starts at 9.8 m/s2 and drops to some constant lower value as the ball approaches the ground.

e) The acceleration is 9.8 m/s2 at all times.

3.3.2. A baseball is hit upward and travels along a parabolic arc before it strikes the ground. Which one of the following statements is necessarily true?

a) The velocity of the ball is a maximum when the ball is at the highest point in the arc.

b) The x-component of the velocity of the ball is the same throughout the ball's flight.

c) The acceleration of the ball decreases as the ball moves upward.

d) The velocity of the ball is zero m/s when the ball is at the highest point in the arc.

e) The acceleration of the ball is zero m/s2 when the ball is at the highest point in the arc.

3.3.2. A baseball is hit upward and travels along a parabolic arc before it strikes the ground. Which one of the following statements is necessarily true?

a) The velocity of the ball is a maximum when the ball is at the highest point in the arc.

b) The x-component of the velocity of the ball is the same throughout the ball's flight.

c) The acceleration of the ball decreases as the ball moves upward.

d) The velocity of the ball is zero m/s when the ball is at the highest point in the arc.

e) The acceleration of the ball is zero m/s2 when the ball is at the highest point in the arc.

3.3.3. Two cannons are mounted on a high cliff. Cannon A fires balls with twice the initial velocity of cannon B. Both cannons are aimed horizontally and fired. How does the horizontal range of cannon A compare to that of cannon B?

a) The range for both balls will be the same

b) The range of the cannon ball B is about 0.7 that of cannon ball A.

c) The range of the cannon ball B is about 1.4 times that of cannon

ball A.

d) The range of the cannon ball B is about 2 times that of cannon ball A.

e) The range of the cannon ball B is about 0.5 that of cannon ball A.

3.3.3. Two cannons are mounted on a high cliff. Cannon A fires balls with twice the initial velocity of cannon B. Both cannons are aimed horizontally and fired. How does the horizontal range of cannon A compare to that of cannon B?

a) The range for both balls will be the same

b) The range of the cannon ball B is about 0.7 that of cannon ball A.

c) The range of the cannon ball B is about 1.4 times that of cannon

ball A.

d) The range of the cannon ball B is about 2 times that of cannon ball A.

e) The range of the cannon ball B is about 0.5 that of cannon ball A.

3.3.4. Which one of the following statements concerning the range of a football is true if the football is kicked at an angle with an initial speed v0?

a) The range is independent of initial speed v0.

b) The range is only dependent on the initial speed v0.

c) The range is independent of the angle.

d) The range is only dependent on the angle.

e) The range is dependent on both the initial speed v0 and the angle.

3.3.4. Which one of the following statements concerning the range of a football is true if the football is kicked at an angle with an initial speed v0?

a) The range is independent of initial speed v0.

b) The range is only dependent on the initial speed v0.

c) The range is independent of the angle.

d) The range is only dependent on the angle.

e) The range is dependent on both the initial speed v0 and the angle.

3.3.5. A bullet is aimed at a target on the wall a distance L away from the firing position. Because of gravity, the bullet strikes the wall a distance

Δy below the mark as suggested in the figure. Note: The drawing is not to scale. If the distance L was half as large, and the bullet had the same

initial velocity, how would Δy be affected?

a) Δy will double.

b) Δy will be half as large.

c) Δy will be one fourth as large.

d) Δy will be four times larger.

e) It is not possible to determine unless numerical values are given for the distances.

3.3.1. A bicyclist is riding at a constant speed along a horizontal, straight-line path. The rider throws a ball straight up to a height a few meters above her head. Ignoring air resistance, where will the ball land?

a) in front of the rider

b) behind the rider

c) in the same hand that threw the ball

d) in the opposite hand to the one that threw it

e) This cannot be determined without knowing the speed of the rider and the maximum height of the ball.

3.3.1. A bicyclist is riding at a constant speed along a horizontal, straight-line path. The rider throws a ball straight up to a height a few meters above her head. Ignoring air resistance, where will the ball land?

a) in front of the rider

b) behind the rider

c) in the same hand that threw the ball

d) in the opposite hand to the one that threw it

e) This cannot be determined without knowing the speed of the rider and the maximum height of the ball.

3.3.2. Football A is kicked at a speed v at an angle of with respect to the horizontal direction. If football B is kicked at the same angle, but with a speed 2v, what is the ratio of the range of B to the range of A?

a) 1

b) 2

c) 3

d) 4

e) 9

3.3.2. Football A is kicked at a speed v at an angle of with respect to the horizontal direction. If football B is kicked at the same angle, but with a speed 2v, what is the ratio of the range of B to the range of A?

a) 1

b) 2

c) 3

d) 4

e) 9

3.3.3. Balls A, B, and C are identical. From the top of a tall building, ball A is launched with a velocity of 20 m/s at an angle of 45 above the horizontal direction, ball B is launched with a velocity of 20 m/s in the horizontal direction, and ball C is launched with a velocity of 20 m/s at an angle of 45 below the horizontal direction. Which of the following choices correctly relates the magnitudes of the velocities of the balls just before they hit the ground below? Ignore any effects of air resistance.

a) vA = vC > vB

b) vA = vC = vB

c) vA > vC > vB

d) vA < vC < vB

e) vA > vC < vB

3.3.3. Balls A, B, and C are identical. From the top of a tall building, ball A is launched with a velocity of 20 m/s at an angle of 45 above the horizontal direction, ball B is launched with a velocity of 20 m/s in the horizontal direction, and ball C is launched with a velocity of 20 m/s at an angle of 45 below the horizontal direction. Which of the following choices correctly relates the magnitudes of the velocities of the balls just before they hit the ground below? Ignore any effects of air resistance.

a) vA = vC > vB

b) vA = vC = vB

c) vA > vC > vB

d) vA < vC < vB

e) vA > vC < vB

3.3.4. A basketball is launched with an initial speed of 8.5 m/s and follows the trajectory shown. The ball enters the basket 0.92 s after it is launched. What are the distances x and y? Note: The drawing is not to scale.

a) x = 6.0 m, y = 0.88 m

b) x = 5.4 m, y = 0.73 m

c) x = 5.7 m, y = 0.91 m

d) x = 7.6 m, y = 1.1 m

e) x = 6.3 m, y = 0.96 m

3.3.4. A basketball is launched with an initial speed of 8.5 m/s and follows the trajectory shown. The ball enters the basket 0.92 s after it is launched. What are the distances x and y? Note: The drawing is not to scale.

a) x = 6.0 m, y = 0.88 m

b) x = 5.4 m, y = 0.73 m

c) x = 5.7 m, y = 0.91 m

d) x = 7.6 m, y = 1.1 m

e) x = 6.3 m, y = 0.96 m

3.3.5. A physics student standing on the edge of a cliff throws a stone vertically downward with an initial speed of 10.0 m/s. The instant before the stone hits the ground below, it is traveling at a speed of 30.0 m/s. If the physics student were to throw the rock horizontally outward from the cliff instead, with the same initial speed of 10.0 m/s, what is the magnitude of the velocity of the stone just before it hits the ground? Ignore any effects of air resistance.

a) 10.0 m/s

b) 20.0 m/s

c) 30.0 m/s

d) 40.0 m/s

e) The height of the cliff must be specified to answer this question.

3.3.5. A physics student standing on the edge of a cliff throws a stone vertically downward with an initial speed of 10.0 m/s. The instant before the stone hits the ground below, it is traveling at a speed of 30.0 m/s. If the physics student were to throw the rock horizontally outward from the cliff instead, with the same initial speed of 10.0 m/s, what is the magnitude of the velocity of the stone just before it hits the ground? Ignore any effects of air resistance.

a) 10.0 m/s

b) 20.0 m/s

c) 30.0 m/s

d) 40.0 m/s

e) The height of the cliff must be specified to answer this question.

3.3.5. At time t = 0 s, Ball A is thrown vertically upward with an initial speed v0A. Ball B is thrown vertically upward shortly after Ball A at time t. Ball B passes Ball A just as Ball A is reaching the top of its trajectory. What is the initial speed v0B of Ball B in terms of the given parameters? The acceleration due to gravity is g.

a) v0B = v0A (1/2)gt2

b) v0B = v0A (1/2)gt

c)

d)

e) v0B = 2v0A gt

gtv

gtvtgvv

A

AAB

0

022

212

00

gtv

tgvv

A

AB

0

2221

00

3.3.5. At time t = 0 s, Ball A is thrown vertically upward with an initial speed v0A. Ball B is thrown vertically upward shortly after Ball A at time t. Ball B passes Ball A just as Ball A is reaching the top of its trajectory. What is the initial speed v0B of Ball B in terms of the given parameters? The acceleration due to gravity is g.

a) v0B = v0A (1/2)gt2

b) v0B = v0A (1/2)gt

c)

d)

e) v0B = 2v0A gt

gtv

gtvtgvv

A

AAB

0

022

212

00

gtv

tgvv

A

AB

0

2221

00

3.3.6. A toy rocket is launched at an angle of 45 with a speed v0. If there is no air resistance, at what point during the time that it is in the air does the speed of the rocket equal 0.5v0?

a) when the rocket is at one half of its maximum height as it is going upward

b) when the rocket is at one half of its maximum height as it is going downward

c) when the rocket is at its maximum height

d) when the rocket is at one fourth of its maximum height as it is going downward

e) at no time during the flight

3.3.6. A toy rocket is launched at an angle of 45 with a speed v0. If there is no air resistance, at what point during the time that it is in the air does the speed of the rocket equal 0.5v0?

a) when the rocket is at one half of its maximum height as it is going upward

b) when the rocket is at one half of its maximum height as it is going downward

c) when the rocket is at its maximum height

d) when the rocket is at one fourth of its maximum height as it is going downward

e) at no time during the flight

3.3.7. During a high school track meet, an athlete performing the long jump runs and leaps at an angle of 25 and lands in a sand pit 8.5 m from his launch point. If the launch point and landing points are at the same height, y = 0 m, with what speed does the athlete land?

a) 6 m/s

b) 8 m/s

c) 10 m/s

d) 2 m/s

e) 4 m/s

3.3.7. During a high school track meet, an athlete performing the long jump runs and leaps at an angle of 25 and lands in a sand pit 8.5 m from his launch point. If the launch point and landing points are at the same height, y = 0 m, with what speed does the athlete land?

a) 6 m/s

b) 8 m/s

c) 10 m/s

d) 2 m/s

e) 4 m/s

3.3.8. An airplane is flying horizontally at a constant velocity when a package is dropped from its cargo bay. Assuming no air resistance, which one of the following statements is correct?

a) The package follows a curved path that lags behind the airplane.

b) The package follows a straight line path that lags behind the airplane.

c) The package follows a straight line path, but it is always vertically below the airplane.

d) The package follows a curved path, but it is always vertically below the airplane.

e) The package follows a curved path, but its horizontal position varies depending on the velocity of the airplane.

3.3.8. An airplane is flying horizontally at a constant velocity when a package is dropped from its cargo bay. Assuming no air resistance, which one of the following statements is correct?

a) The package follows a curved path that lags behind the airplane.

b) The package follows a straight line path that lags behind the airplane.

c) The package follows a straight line path, but it is always vertically below the airplane.

d) The package follows a curved path, but it is always vertically below the airplane.

e) The package follows a curved path, but its horizontal position varies depending on the velocity of the airplane.

3.3.9. In making a movie, a stuntman has to jump from one roof onto another roof, located 2.0 m below. The buildings are separated by a distance of 2.5 m. What is the minimum horizontal speed that the stuntman must have when jumping from the first roof to have a successful jump?

a) 3.9 m/s

b) 2.5 m/s

c) 4.3 m/s

d) 4.5 m/s

e) 3.1 m/s

3.3.9. In making a movie, a stuntman has to jump from one roof onto another roof, located 2.0 m below. The buildings are separated by a distance of 2.5 m. What is the minimum horizontal speed that the stuntman must have when jumping from the first roof to have a successful jump?

a) 3.9 m/s

b) 2.5 m/s

c) 4.3 m/s

d) 4.5 m/s

e) 3.1 m/s

3.3.10. When a projectile is launched at an angle from a height h1 and the projectile lands at the same height, the maximum range, in the absence of air resistance, occurs when = 45. The same projectile is then launched at an angle from a height h1, but it lands at a height h2 that is higher than h1, but less than the maximum height reached by the projectile when = 45. In this case, in the absence of air resistance, does the maximum range still occur for = 45? All angles are measured with respect to the horizontal direction.

a) Yes, = 45 will always have longest range regardless of the height h2.

b) No, depending on the height h2, the longest range may be reached for angles less than 45.

c) No, depending on the height h2, the longest range may be reached for angles greater than 45.

3.3.10. When a projectile is launched at an angle from a height h1 and the projectile lands at the same height, the maximum range, in the absence of air resistance, occurs when = 45. The same projectile is then launched at an angle from a height h1, but it lands at a height h2 that is higher than h1, but less than the maximum height reached by the projectile when = 45. In this case, in the absence of air resistance, does the maximum range still occur for = 45? All angles are measured with respect to the horizontal direction.

a) Yes, = 45 will always have longest range regardless of the height h2.

b) No, depending on the height h2, the longest range may be reached for angles less than 45.

c) No, depending on the height h2, the longest range may be reached for angles greater than 45.

3.3.11. Packages A and B are dropped from the same height simultaneously. Package A is dropped from an airplane that is flying due east at constant speed. Package B is dropped from rest from a helicopter hovering in a stationary position above the ground. Ignoring air friction effects, which of the following statements is true?

a) A and B reach the ground at the same time, but B has a greater velocity in the vertical direction.

b) A and B reach the ground at the same time; and they have the same velocity in the vertical direction.

c) A and B reach the ground at different times because B has a greater velocity in both the horizontal and vertical directions.

d) A and B reach the ground at different times; and they have the same velocity in the vertical direction.

e) A reaches the ground first because it falls straight down, while B has to travel much further than A.

3.3.11. Packages A and B are dropped from the same height simultaneously. Package A is dropped from an airplane that is flying due east at constant speed. Package B is dropped from rest from a helicopter hovering in a stationary position above the ground. Ignoring air friction effects, which of the following statements is true?

a) A and B reach the ground at the same time, but B has a greater velocity in the vertical direction.

b) A and B reach the ground at the same time; and they have the same velocity in the vertical direction.

c) A and B reach the ground at different times because B has a greater velocity in both the horizontal and vertical directions.

d) A and B reach the ground at different times; and they have the same velocity in the vertical direction.

e) A reaches the ground first because it falls straight down, while B has to travel much further than A.

3.3.5. A bullet is aimed at a target on the wall a distance L away from the firing position. Because of gravity, the bullet strikes the wall a distance

Δy below the mark as suggested in the figure. Note: The drawing is not to scale. If the distance L was half as large, and the bullet had the same

initial velocity, how would Δy be affected?

a) Δy will double.

b) Δy will be half as large.

c) Δy will be one fourth as large.

d) Δy will be four times larger.

e) It is not possible to determine unless numerical values are given for the distances.

3.4 Relative Velocity

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Example 11 Crossing a River

The engine of a boat drives it across a river that is 1800m wide.The velocity of the boat relative to the water is 4.0m/s directed perpendicular to the current. The velocity of the water relativeto the shore is 2.0m/s.

(a) What is the velocity of the boat relative to the shore?

(b) How long does it take for the boat to cross the river?


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s 450sm4.0

m 1800t

3.4.1. At an air show, three planes are flying horizontally due east. The velocity of plane A relative to plane B is vAB; the velocity of plane A relative to plane C is vAC; and the velocity of plane B relative to plane C is vBC. Determine vAB if vAC = +10 m/s and vBC = +20 m/s?

a) 10 m/s

b) +10 m/s

c) 20 m/s

d) +20 m/s

e) zero m/s

3.4.1. At an air show, three planes are flying horizontally due east. The velocity of plane A relative to plane B is vAB; the velocity of plane A relative to plane C is vAC; and the velocity of plane B relative to plane C is vBC. Determine vAB if vAC = +10 m/s and vBC = +20 m/s?

a) 10 m/s

b) +10 m/s

c) 20 m/s

d) +20 m/s

e) zero m/s

3.4.2. A train is traveling due east at a speed of 26.8 m/s relative to the ground. A passenger is walking toward the front of the train at a speed of 1.7 m/s relative to the train. Directly overhead the train is a plane flying horizontally due west at a speed of 257.0 m/s relative to the ground. What is the horizontal component of the velocity of the airplane with respect to the passenger on the train?

a) 258.7 m/s, due west

b) 285.5 m/s, due west

c) 226.8 m/s, due west

d) 231.9 m/s, due west

e) 257.0 m/s, due west

3.4.2. A train is traveling due east at a speed of 26.8 m/s relative to the ground. A passenger is walking toward the front of the train at a speed of 1.7 m/s relative to the train. Directly overhead the train is a plane flying horizontally due west at a speed of 257.0 m/s relative to the ground. What is the horizontal component of the velocity of the airplane with respect to the passenger on the train?

a) 258.7 m/s, due west

b) 285.5 m/s, due west

c) 226.8 m/s, due west

d) 231.9 m/s, due west

e) 257.0 m/s, due west

3.4.3. Sailors are throwing a football on the deck of an aircraft carrier as it is sailing with a constant velocity due east. Sailor A is standing on the west side of the flight deck while sailor B is standing on the east side. Sailors on the deck of another aircraft carrier that is stationary are watching the football as it is being tossed back and forth as the first carrier passes. Assume that sailors A and B throw the football with the same initial speed at the same launch angle with respect to the horizontal, do the sailors on the stationary carrier see the football follow the same parabolic trajectory as the ball goes east to west as it does when it goes west to east?

a) Yes, to the stationary sailors, the trajectory the ball follows is the same whether it is traveling west to east or east to west.

b) No, to the stationary sailors, the length of the trajectory appears shorter as it travels west to east than when it travels east to west.

c) No, to the stationary sailors, the ball appears to be in the air for a much longer time when it is traveling west to east than when it travels east to west.

d) No, to the stationary sailors, the length of the trajectory appears longer as it travels west to east than when it travels east to west.

e) No, to the stationary sailors, the ball appears to be in the air for a much shorter time when it is traveling west to east than when it travels east to west.

3.4.3. Sailors are throwing a football on the deck of an aircraft carrier as it is sailing with a constant velocity due east. Sailor A is standing on the west side of the flight deck while sailor B is standing on the east side. Sailors on the deck of another aircraft carrier that is stationary are watching the football as it is being tossed back and forth as the first carrier passes. Assume that sailors A and B throw the football with the same initial speed at the same launch angle with respect to the horizontal, do the sailors on the stationary carrier see the football follow the same parabolic trajectory as the ball goes east to west as it does when it goes west to east?

a) Yes, to the stationary sailors, the trajectory the ball follows is the same whether it is traveling west to east or east to west.

b) No, to the stationary sailors, the length of the trajectory appears shorter as it travels west to east than when it travels east to west.

c) No, to the stationary sailors, the ball appears to be in the air for a much longer time when it is traveling west to east than when it travels east to west.

d) No, to the stationary sailors, the length of the trajectory appears longer as it travels west to east than when it travels east to west.

e) No, to the stationary sailors, the ball appears to be in the air for a much shorter time when it is traveling west to east than when it travels east to west.

Kinematics in Two Dimensions Chapter 3. 3.1 Displacement, Velocity, and Acceleration.

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