4
CHAPTER OUTLINE
4.1
The Position, Velocity, and
Acceleration Vectors
4.2
TwoDimensional Motion
with Constant Acceleration
4.3
Projectile Motion
4.4
Uniform Circular Motion
4.5
Tangential and Radial
Acceleration
4.6
Relative Velocity and
Relative Acceleration
Motion in Two Dimensions
ANSWERS TO QUESTIONS
Q4.1
Yes. An object moving in uniform circular motion moves at a
constant speed, but changes its direction of motion. An object
cannot accelerate if its velocity is constant.
Q4.2
No, you cannot determine the instantaneous velocity. Yes, you
can determine the average velocity. The points could be widely
separated. In this case, you can only determine the average
velocity, which is
v
x
=
∆
∆
t
.
Q4.3
(a)
a
a
a
a
v
v
v
v
(b)
a
a
v
v
a
v
a
v
a
v
Q4.4
(a)
10 m s
±
i
(b)
−
980
.
m
s
2
±
j
Q4.5
The easiest way to approach this problem is to determine acceleration first, velocity second and
finally position.
Vertical: In free flight,
ag
y
=−
. At the top of a projectile’s trajectory,
v
y
=
0. Using this, the
maximum height can be found using
vv
a
y
y
fy
iy
y
f
i
22
2
=+
−
di
.
Horizontal:
a
x
=
0 , so
v
x
is always the same. To find the horizontal position at maximum
height, one needs the flight time,
t
. Using the vertical information found previously, the flight time
can be found using
a
t
fy
iy
y
. The horizontal position is
xv
t
fi
x
=
.
If air resistance is taken into account, then the acceleration in both the
x
and
y
directions
would have an additional term due to the drag.
Q4.6
A parabola.
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Motion in Two Dimensions
Q4.7
The balls will be closest together as the second ball is thrown. Yes, the first ball will always be
moving faster, since its flight time is larger, and thus the vertical component of the velocity is larger.
The time interval will be one second. No, since the vertical component of the motion determines the
flight time.
Q4.8
The ball will have the greater speed. Both the rock and the ball will have the same vertical
component of the velocity, but the ball will have the additional horizontal component.
Q4.9
(a)
yes
(b)
no
(c)
no
(d)
yes
(e)
no
Q4.10
Straight up. Throwing the ball any other direction than straight up will give a nonzero speed at the
top of the trajectory.
Q4.11
No. The projectile with the larger vertical component of the initial velocity will be in the air longer.
Q4.12
The projectile is in free fall. Its vertical component of acceleration is the downward acceleration of
gravity. Its horizontal component of acceleration is zero.
Q4.13
(a)
no
(b)
yes
(c)
yes
(d)
no
Q4.14
60
°
. The projection angle appears in the expression for horizontal range in the function sin2
θ
.
This
function is the same for 30
°
and 60
°
.
Q4.15
The optimal angle would be less than 45
°
. The longer the projectile is in the air, the more that air
resistance will change the components of the velocity. Since the vertical component of the motion
determines the flight time, an angle less than 45
°
would increase range.
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 Spring '08
 Hadley
 Acceleration, Circular Motion, Projectile Motion

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