98
v
(m/s)0
5
–5
t (s)
0510
Problem 19.
Problem 20.
Chapter 3Acceleration and Free Fall
19 S. The graph represents the motion of a rolling ball that bounces off of a
wall. When does the ball return to the location it had at t=0.
20 S. (a) The ball is released at the top of the ramp shown in the figure.
Friction is negligible. Use physical reasoning to draw v-t and a-t graphs.
Assume that the ball doesn’t bounce at the point where the ramp changes
slope. (b) Do the same for the case where the ball is rolled up the slope from
the right side, but doesn’t quite have enough speed to make it over the top.
21 S. You throw a rubber ball up, and it falls and bounces several times.
Draw graphs of position, velocity, and acceleration as functions of time.
22 S. Starting from rest, a ball rolls down a ramp, traveling a distance L and
picking up a final speed v. How much of the distance did the ball have to
cover before achieving a speed of v/2. [Based on a problem by Arnold
Arons.]
23. The graph shows the acceleration of a chipmunk in a TV cartoon. It
consists of two circular arcs and two line segments. At t=0, the chipmunk’s
velocity is –3.1 m/s. What is its velocity at t=10 s.
24. Find the error in the following calculation. A student wants to find the
distance traveled by a car that accelerates from rest for 5.0 s with an accel-
eration of 2.0 m/s
2
. First he solves a=
.
v/
.
t for
.
v=10 m/s. Then he
multiplies to find (10 m/s)(5.0 s)=50 m. Do not just recalculate the result
by a different method; if that was all you did, you’d have no way of knowing
which calculation was correct, yours or his.
25. Acceleration could be defined either as
.
v/
.
t or as the slope of the
tangent line on the v-t graph. Is either one superior as a definition, or are
they equivalent. If you say one is better, give an example of a situation
where it makes a difference which one you use.
26. If an object starts accelerating from rest, we have v
2
=2a
.
x for its speed
after it has traveled a distance
.
x. Explain in words why it makes sense that
the equation has velocity squared, but distance only to the first power.
Don’t recapitulate the derivation in the book, or give a justification based
on units. The point is to explain what this feature of the equation tells us
about the way speed increases as more distance is covered.
27. The figure shows a practical, simple experiment for determining g to
high precision. Two steel balls are suspended from electromagnets, and are
released simultaneously when the electric current is shut off. They fall
through unequal heights
.
x
1
and
.
x
2
. A computer records the sounds
through a microphone as first one ball and then the other strikes the floor.
From this recording, we can accurately determine the quantity T defined as
T=
.
t
2
-
.
t
1
, i.e., the time lag between the first and second impacts. Note
that since the balls do not make any sound when they are released, we have
no way of measuring the individual times
.
t
2
and
.
t
1
. (a) Find an
equation for g in terms of the measured quantities T,
.
x
1
and
.
x
2
. (b)
Check the units of your equation. (c) Check that your equation gives the
correct result in the case where
.
x
1
=0. However, is this case realistic. (d)
What happens when
.
x
1
=
.
x
2
.
a
(m/s
2
)
10
0
t (s)
0510
Problem 23.
Problem 27.