51
Motion in Two Dimensions
CHAPTER OUTLINE
3.1
The Position, Velocity,
and Acceleration Vectors
3.2
TwoDimensional Motion
with Constant
Acceleration
3.3
Projectile Motion
3.4
The Particle in Uniform
Circular Motion
3.5
Tangential and Radial
Acceleration
3.6
Relative Velocity
3.7
Context
Connection
Lateral
Acceleration of
Automobiles
ANSWERS TO QUESTIONS
Q3.1
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
r
r
v
x
=
∆
∆
t
.
Q3.2
(a)
r
v
r
v
r
v
r
v
r
a
r
a
r
a
r
a
FIG. Q3.2(a)
(b)
r
v
r
v
r
v
r
v
r
a
r
a
r
a
r
a
r
v
r
a
FIG. Q3.2(b)
Q3.3
The easiest way to approach this problem is to determine acceleration first, velocity second and
finally position. Determine
(c)
acceleration first,
(b)
velocity second, and
(a)
finally position.
Vertical: In free flight,
a
g
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 were taken into account, then the acceleration in both the
x
and
y
directions
would have an additional term due to the drag.
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Motion in Two Dimensions
Q3.4
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.
Q3.5
A parabola.
Q3.6
(a)
no
(b)
yes
(c)
yes
(d)
no
Q3.7
The projectile is in free fall. Its vertical component of acceleration is the downward acceleration of
gravity. Its horizontal component of acceleration is zero.
Q3.8
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.
Q3.9
The projectile on the moon would have both the larger range and the greater altitude.
Apollo
astronauts performed the experiment with golf balls.
Q3.10
The racing car rounds the turn at a constant
speed
of 90 miles per hour.
Q3.11
(a)
no
(b)
yes
In the second case, the particle is continuously changing the direction of its velocity vector.
Q3.12
(a)
The velocity is not constant because the object is constantly changing the direction of its
motion.
(b)
The acceleration is not constant because the acceleration always points towards the center of
the circle. The magnitude of the acceleration is constant, but not the direction.
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 Spring '08
 DAFSF
 Acceleration, Circular Motion, Projectile Motion, Velocity, Cos, Yi, Two Dimensions, tan 1

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