The graph represents the velocity of a bee along a straight line.
At t = 0, the bee is at the hive. (a) When is the bee farthest from
the hive? (b) How far is the bee at its farthest point from the hive?
(c) At t = 13s, how far is the bee from the hive? [Hint: Try problem
A rock is dropped into a pond. Draw plots of its position versus
time, velocity versus time, and acceleration versus time. Include its
whole motion, starting from the moment it is dropped, and continuing
while it falls through the air, passes through the water, and ends
up at rest on the bottom of the pond. Do your work on photocopy
or a printout of page 122.
Legende zu Bild
In an 18th-century naval battle, a cannon ball is shot horizontally,
passes through the side of an enemy ship's hull, flies across the
galley, and lodges in a bulkhead. Draw plots of its horizontal position,
velocity, and acceleration as functions of time, starting while it
is inside the cannon and has not yet been fired, and ending when it
comes to rest. There is not any significant amount of friction from
the air. Although the ball may rise and fall, you are only concerned
with its horizontal motion, as seen from above. Do your work on
photocopy or a printout of page 122.
Draw graphs of position, velocity, and acceleration as functions
of time for a person bunjee jumping. (In bunjee jumping, a person
has a stretchy elastic cord tied to his/her ankles, and jumps off of a
high platform. At the bottom of the fall, the cord brings the person
up short. Presumably the person bounces up a little.) Do your work
on photocopy or a printout of page 122.
Legende zu Bild
A ball rolls down the ramp shown in the figure, consisting of a
curved knee, a straight slope, and a curved bottom. For each part of
the ramp, tell whether the ball's velocity is increasing, decreasing,
or constant, and also whether the ball's acceleration is increasing,
decreasing, or constant. Explain your answers. Assume there is no
air friction or rolling resistance. Hint: Try problem 20 first. [Based
on a problem by Hewitt.]
A toy car is released on one side of a piece of track that is bent
into an upright U shape. The car goes back and forth. When the
car reaches the limit of its motion on one side, its velocity is zero.
Is its acceleration also zero? Explain using a v -t graph. [Based on
a problem by Serway and Faughn.]
What is the acceleration of a car that moves at a steady velocity
of 100 km/h for 100 seconds? Explain your answer. [Based on a
problem by Hewitt.]
A physics homework question asks, "If you start from rest and
accelerate at 1.54 m/s2 for 3.29 s, how far do you travel by the end
of that time?" A student answers as follows:
1.54 × 3.29 = 5.07 m
His Aunt Wanda is good with numbers, but has never taken physics.
She doesn't know the formula for the distance traveled under constant
acceleration over a given amount of time, but she tells her
nephew his answer cannot be right. How does she know?
You are looking into a deep well. It is dark, and you cannot
see the bottom. You want to find out how deep it is, so you drop
a rock in, and you hear a splash 3.0 seconds later. How deep is the
You take a trip in your spaceship to another star. Setting off,
you increase your speed at a constant acceleration. Once you get
half-way there, you start decelerating, at the same rate, so that by
the time you get there, you have slowed down to zero speed. You see
the tourist attractions, and then head home by the same method.
(a) Find a formula for the time, T, required for the round trip, in
terms of d, the distance from our sun to the star, and a, the magnitude
of the acceleration. Note that the acceleration is not constant
over the whole trip, but the trip can be broken up into constantacceleration
(b) The nearest star to the Earth (other than our own sun) is Proxima
Centauri, at a distance of d = 4 × 1016m. Suppose you use an
acceleration of a = 10 m/s2, just enough to compensate for the lack
of true gravity and make you feel comfortable. How long does the
round trip take, in years?
(c) Using the same numbers for d and a, find your maximum speed.
Compare this to the speed of light, which is 3.0 × 108 m/s. (Later
in this course, you will learn that there are some new things going
on in physics when one gets close to the speed of light, and that it
is impossible to exceed the speed of light. For now, though, just use
the simpler ideas you've learned so far.)
You climb half-way up a tree, and drop a rock. Then you
climb to the top, and drop another rock. How many times greater
is the velocity of the second rock on impact? Explain. (The answer
is not two times greater.)
Alice drops a rock off a cliff. Bubba shoots a gun straight
down from the edge of the same cliff. Compare the accelerations of
the rock and the bullet while they are in the air on the way down.
[Based on a problem by Serway and Faughn.]
A person is parachute jumping. During the time between
when she leaps out of the plane and when she opens her chute, her
altitude is given by an equation of the form
y = b - c (t + ke-t/k) ,
where e is the base of natural logarithms, and b, c, and k are constants.
Because of air resistance, her velocity does not increase at a
steady rate as it would for an object falling in vacuum.
(a) What units would b, c, and k have to have for the equation to
(b) Find the person's velocity, v, as a function of time. [You will
need to use the chain rule, and the fact that d(ex)/dx = ex.]
(c) Use your answer from part (b) to get an interpretation of the
constant c. [Hint: e-x approaches zero for large values of x.]
(d) Find the person's acceleration, a, as a function of time.
(e) Use your answer from part (b) to show that if she waits long
enough to open her chute, her acceleration will become very small.
The top part of the figure shows the position-versus-time
graph for an object moving in one dimension. On the bottom part
of the figure, sketch the corresponding v-versus-t graph.
Solution, p. 276
On New Year's Eve, a stupid person fires a pistol straight up.
The bullet leaves the gun at a speed of 100 m/s. How long does it
take before the bullet hits the ground?
Solution, p. 276
If the acceleration of gravity on Mars is 1/3 that on Earth,
how many times longer does it take for a rock to drop the same
distance on Mars? Ignore air resistance.
Solution, p. 277
A honeybee's position as a function of time is given by
x = 10t - t3, where t is in seconds and x in meters. What is its
acceleration at t = 3.0 s?
Solution, p. 277 ∫
In July 1999, Popular Mechanics carried out tests to find
which car sold by a major auto maker could cover a quarter mile
(402 meters) in the shortest time, starting from rest. Because the
distance is so short, this type of test is designed mainly to favor the
car with the greatest acceleration, not the greatest maximum speed
(which is irrelevant to the average person). The winner was the
Dodge Viper, with a time of 12.08 s. The car's top (and presumably
final) speed was 118.51 miles per hour (52.98 m/s). (a) If a car,
starting from rest and moving with constant acceleration, covers
a quarter mile in this time interval, what is its acceleration? (b)
What would be the final speed of a car that covered a quarter mile
with the constant acceleration you found in part a? (c) Based on
the discrepancy between your answer in part b and the actual final
speed of the Viper, what do you conclude about how its acceleration
changed over time?
Solution, p. 277
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?
Solution, p. 277
(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.
Solution, p. 277
You throw a rubber ball up, and it falls and bounces several
times. Draw graphs of position, velocity, and acceleration as
functions of time.
Solution, p. 278
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.]
Solution, p. 279
The graph shows the acceleration of a chipmunk in a TV
cartoon. It consists of two circular arcs and two line segments.
At t = 0.00 s, the chipmunk's velocity is -3.10 m/s. What is its
velocity at t = 10.00 s?
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 acceleration of 2.0 m/s2. First he solves a = Δv/Δt for
Δv = 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.
Acceleration could be defined either as Δv/Δt 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
If an object starts accelerating from rest, we have v2 = 2aΔx
for its speed after it has traveled a distance Δx. 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.
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 Δx1 and
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 = Δt2
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 Δt2 and Δt1.
(a) Find an equation for g in terms of the measured quantities T,
Δx1 and Δx2.
(b) Check the units of your equation.
(c) Check that your equation gives the correct result in the case
where Δx1 is very close to zero. However, is this case realistic?
(d) What happens when Δx1 = Δx2? Discuss this both mathematically
The speed required for a low-earth orbit is 7.9×103 m/s(see
ch. 10). When a rocket is launched into orbit, it goes up a little at
first to get above almost all of the atmosphere, but then tips over
horizontally to build up to orbital speed. Suppose the horizontal
acceleration is limited to 3g to keep from damaging the cargo (or
hurting the crew, for a crewed flight). (a) What is the minimum
distance the rocket must travel downrange before it reaches orbital
speed? How much does it matter whether you take into account the
initial eastward velocity due to the rotation of the earth? (b) Rather
than a rocket ship, it might be advantageous to use a railgun design,
in which the craft would be accelerated to orbital speeds along a
railroad track. This has the advantage that it isn't necessary to lift
a large mass of fuel, since the energy source is external. Based on
your answer to part a, comment on the feasibility of this design for
crewed launches from the earth's surface.
Some fleas can jump as high as 30 cm. The flea only has a
short time to build up speed - the time during which its center of
mass is accelerating upward but its feet are still in contact with the
ground. Make an order-of-magnitude estimate of the acceleration
the flea needs to have while straightening its legs, and state your
answer in units of g, i.e., how many "g's it pulls." (For comparison,
fighter pilots black out or die if they exceed about 5 or 10 g's.)
Consider the following passage from Alice in Wonderland, in
which Alice has been falling for a long time down a rabbit hole:
Down, down, down. Would the fall never come to an end? "I
wonder how many miles I've fallen by this time?" she said aloud.
"I must be getting somewhere near the center of the earth. Let me
see: that would be four thousand miles down, I think" (for, you see,
Alice had learned several things of this sort in her lessons in the
schoolroom, and though this was not a very good opportunity for
showing off her knowledge, as there was no one to listen to her, still
it was good practice to say it over)...
Alice doesn't know much physics, but let's try to calculate the
amount of time it would take to fall four thousand miles, starting
from rest with an acceleration of 10 m/s2. This is really only a lower
limit; if there really was a hole that deep, the fall would actually
take a longer time than the one you calculate, both because there
is air friction and because gravity gets weaker as you get deeper (at
the center of the earth, g is zero, because the earth is pulling you
equally in every direction at once).