CHAPTER
3
KINEMATICS IN
TWO DIMENSIONS
ANSWERS TO FOCUS ON CONCEPTS QUESTIONS
1.
(a)
The horizontal component
v
x
of the projectile’s velocity remains constant throughout the
motion, since the acceleration
a
x
in the horizontal direction is zero (
a
x
= 0 m/s
2
). The
vertical component
v
y
, however, changes as the projectile moves. This component is greatest
at point 1, decreases to zero at point 2 at the top of the trajectory, and then increases to a
magnitude less than that at point 1 as the projectile approaches point 3.
2.
(b)
The minimum speed of the projectile occurs when it is at the top of its trajectory. At this
point the vertical component of its velocity is zero (
v
y
= 0 m/s). Since there is no
acceleration in the
x
direction (
a
x
= 0 m/s
2
), the
x
component of the projectile’s velocity
remains constant at
v
x
=
+30 m/s throughout the motion. Thus, the minimum speed is
30 m/s.
3.
(c)
The acceleration due to gravity is the same for both balls, despite the fact that they have
different velocities.
4.
(d)
The acceleration of a projectile is the same at all points on the trajectory. It points
downward, toward the earth, and has a magnitude of 9.80 m/s
2
.
5.
(c)
Since there is no acceleration in the
x
direction (
a
x
= 0 m/s
2
) for projectile motion, the
x
component
v
x
of the velocity is constant throughout the motion. And, the acceleration due to
gravity,
a
y
=
−
9.80 m/s
2
(“downward” is the negative direction), also remains constant.
6.
(c)
The time for a projectile to reach the ground depends only on the
y
component (or
vertical component) of its variables, i.e.,
y
,
v
0
y
, and
a
y
. These variables are the same for both
balls. The fact that Ball 1 is moving horizontally at the top of its trajectory does not play a
role in the time it takes for it to reach the ground.
7.
(a)
Using
y
=
−
19.6 m,
a
y
=
−
9.80 m/s
2
, and
v
0
y
= 0 m/s, Equation 3.5b
(
)
2
1
0
2
y
y
y
v
t
a t
=
+
can be used to calculate the time
t
.
8.
(b)
The time a projectile is in the air is equal to twice the time it takes to fall from its
maximum height. Both have the same maximum height, so the time of fall is the same.
Therefore, both projectiles are in the air for the same amount of time.

98
KINEMATICS IN TWO DIMENSIONS
9.
(a)
The time a projectile is in the air is equal to twice the time it takes to fall from its
maximum height. Projectile 1 reaches the greater height, so it spends the greater amount of
time in the air.
10. (c)
The vertical component (or
y
component) of the final velocity depends on the
y
components of the kinematic variables (
y
,
v
0
y
, and
a
y
) and the time
t
. These variables are the
same for both balls, so they have the same vertical component of the velocity.
11. (a)
A person standing on the ground sees a car traveling at +25 m/s. When the driver of the
car looks out the window, she sees the person moving in the opposite direction with the
same speed, or with a velocity of
−
25 m/s.

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