Chapter 3
Kinematics in Two Dimensions
3.1 Displacement, Velocity, and Acceleration

ro  initialposition

r  finalposition
  
r  r  ro  displacement
3.1 Displacement, Velocity, and Acceleration
Average velocity is the
displacement divided by
the elapsed time.
 

 r  ro r
v

t  to t
3.1 Displacement, Velocity, and Acceleration
The instantaneous velocity indicates how fast
the car moves and the direction of motion at each
instant of time.


r
v  lim
t 0 t
3.1 Displacement, Velocity, and Acceleration


r
v  lim
t 0 t
3.1 Displacement, Velocity, and Acceleration
DEFINITION OF AVERAGE ACCELERATION
 

 v  v o v
a

t  to
t

v

v

vo
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
b) instantaneous velocity
c) average displacement
d) average acceleration
e) instantaneous acceleration
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
b) instantaneous velocity
c) average displacement
d) average acceleration
e) instantaneous acceleration
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
b) instantaneous velocity
c) average displacement
d) average acceleration
e) instantaneous acceleration
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
v  vo  at
x
1
2
vo  v t
v  v  2ax
2
2
o
x  vot  at
1
2
2
3.2 Equations of Kinematics in Two Dimensions
x  12 vox  vx  t
vx  vox  axt
x  voxt  axt
1
2
2
v  v  2ax x
2
x
2
ox
3.2 Equations of Kinematics in Two Dimensions
vy  voy  ayt
y  voyt  ayt
1
2
2
y  12 voy  vy  t
v  v  2ay y
2
y
2
oy
3.2 Equations of Kinematics in Two Dimensions
The x part of the motion occurs exactly as it would if the
y part did not occur at all, and vice versa.
3.2 Equations of Kinematics in Two Dimensions
Example 1 A Moving Spacecraft
In the x direction, the spacecraft has an initial velocity component
of +22 m/s and an acceleration of +24 m/s2. In the y direction, the
analogous 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 the
spacecraft at time 7.0 s.
3.2 Equations of Kinematics in Two Dimensions
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 associated with each direction.
4. Verify that the information contains values for at least three
of the kinematic variables. Do this for x and y. 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.
3.2 Equations of Kinematics in Two Dimensions
Example 1 A Moving Spacecraft
In the x direction, the spacecraft has an initial velocity component
of +22 m/s and an acceleration of +24 m/s2. In the y direction, the
analogous 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 the
spacecraft 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
3.2 Equations of Kinematics in Two Dimensions
x
ax
vx
vox
t
?
+24.0 m/s2
?
+22 m/s
7.0 s
x  vox t  a xt
1
2
2
 22 m s 7.0 s  
1
2
vx  vox  axt

24 m s 7.0 s
2

2
 740 m
 22 m s  24 m s 7.0 s   190m s
2
3.2 Equations of Kinematics in Two Dimensions
y
ay
vy
voy
t
?
+12.0 m/s2
?
+14 m/s
7.0 s
y  voy t  12 a y t 2

 14 m s 7.0 s   12 m s
1
2
v y  voy  a y t


2
7.0 s
2
 390 m
 14 m s   12 m s 7.0 s   98m s
2
3.2 Equations of Kinematics in Two Dimensions
v
vy  98m s

vx  190m s
v
190 m s 
2
 98 m s   210 m s
2
  tan 98 190  27
1

3.2 Equations of Kinematics in Two Dimensions
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
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
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.
ay  9.80m s
2
ax  0
vx  vox  constant
3.3 Projectile Motion
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 required
for the care package to hit the ground.
3.3 Projectile Motion
y
ay
-1050 m -9.80 m/s2
vy
voy
t
0 m/s
?
3.3 Projectile Motion
y
ay
vy
-1050 m -9.80 m/s2
y  voyt  ayt
1
2
t
2y

ay
2
voy
t
0 m/s
?
y  ayt
1
2
2
2 1050 m 

14
.
6
s
2
 9.80 m s
3.3 Projectile Motion
Example 4 The Velocity of the Care Package
What are the magnitude and direction of the final velocity of
the care package?
3.3 Projectile Motion
y
ay
-1050 m -9.80 m/s2
vy
voy
t
?
0 m/s
14.6 s
3.3 Projectile Motion
y
ay
vy
voy
t
?
0 m/s
14.6 s
-1050 m -9.80 m/s2


v y  voy  a y t  0   9.80 m s 14.6 s 
 143m s
2
3.3 Projectile Motion
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 point
a rifle straight up into the air and fire it. In the absence of air
resistance, where would the bullet land – behind you, ahead
of you, or in the barrel of the rifle?
3.3 Projectile Motion
Example 6 The Height of a Kickoff
A placekicker kicks a football at and angle of 40.0 degrees and
the initial speed of the ball is 22 m/s. Ignoring air resistance,
determine the maximum height that the ball attains.
3.3 Projectile Motion
vo

voy
vox
voy  vo sin   22m ssin 40  14m s

vox  vo sin   22m scos40  17m s

3.3 Projectile Motion
y
ay
vy
voy
?
-9.80 m/s2
0
14 m/s
t
3.3 Projectile Motion
y
ay
vy
voy
?
-9.80 m/s2
0
14 m/s
v  v  2ay y
2
y
2
oy
y
t
v v
2
y
2a y
0  14 m s 
y


10
m
2
2  9.8 m s
2


2
oy
3.3 Projectile Motion
Example 7 The Time of Flight of a Kickoff
What is the time of flight between kickoff and landing?
3.3 Projectile Motion
y
ay
0
-9.80 m/s2
vy
voy
t
14 m/s
?
3.3 Projectile Motion
y
ay
vy
0
-9.80 m/s2
voy
t
14 m/s
?
y  voyt  ayt
1
2

2

0  14m st   9.80m s t
1
2

2
2

0  214m s   9.80m s t
t  2.9 s
2
3.3 Projectile Motion
Example 8 The Range of a Kickoff
Calculate the range R of the projectile.
x  vox t  a x t  vox t
1
2
2
 17 m s 2.9 s   49 m
3.3 Projectile Motion
Conceptual Example 10
Two Ways to Throw a Stone
From the top of a cliff, a person throws two stones. The stones
have identical initial speeds, but stone 1 is thrown downward
at some angle above the horizontal and stone 2 is thrown at
the 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.
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.
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)
v0 B 
d) v0 B 
v02A  12 g 2t 2  v0 A gt
v0 A  gt
v0 A  12 g 2t 2
v0 A  gt
e) v0B = 2v0A  gt
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)
v0 B 
d) v0 B 
v02A  12 g 2t 2  v0 A gt
v0 A  gt
v0 A  12 g 2t 2
v0 A  gt
e) v0B = 2v0A  gt
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
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
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



v PG  v PT  vTG
3.4 Relative Velocity
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 relative
to 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?
3.4 Relative Velocity



vBS  vBW  v WS
 4.0 

  tan 

63

 2.0 
1
vBS  v
2
BW
 4.5 m s
v
2
WS

4.0 m s   2.0 m s 
2
2
3.4 Relative Velocity
1800m
t
 450s
4.0m 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
3.4.1. 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.