The graph shows the motion of a car stuck in stop-and-go
freeway traffic. (a) If you only knew how far the car had gone
during this entire time period, what would you think its velocity
was? (b) What is the car's maximum velocity?
(a) Let θ be the latitude of a point on the Earth's surface.
Derive an algebra equation for the distance, L, traveled by that
point during one rotation of the Earth about its axis, i.e. over one
day, expressed in terms of L, θ, and R, the radius of the earth.
Check: Your equation should give L = 0 for the North Pole.
(b) At what speed is Fullerton, at latitude θ = 34°, moving with
the rotation of the Earth about its axis? Give your answer in units
of mi/h. [See the table in the back of the book for the relevant
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 the equation
y = (10000 m) - (50 m/s) [t + (5.0 s)e-t/5.0 s] .
Find her velocity at t = 7.0 s. (This can be done on a calculator,
without knowing calculus.) Because of air resistance, her velocity
does not increase at a steady rate as it would for an object falling
A light-year is a unit of distance used in astronomy, and defined
as the distance light travels in one year. The speed of light is 3.0×108
m/s. Find how many meters there are in one light-year, expressing
your answer in scientific notation.
Solution, p. 276
You're standing in a freight train, and have no way to see out.
If you have to lean to stay on your feet, what, if anything, does that
tell you about the train's velocity? Explain.
Solution, p. 276
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 velocity
at t = 3.0 s?
The figure shows the motion of a point on the rim of a rolling
wheel. (The shape is called a cycloid.) Suppose bug A is riding on
the rim of the wheel on a bicycle that is rolling, while bug B is on
the spinning wheel of a bike that is sitting upside down on the floor.
Bug A is moving along a cycloid, while bug B is moving in a circle.
Both wheels are doing the same number of revolutions per minute.
Which bug has a harder time holding on, or do they find it equally
Solution, p. 276
Peanut plants fold up their leaves at night. Estimate the top
speed of the tip of one of the leaves shown in the figure, expressing
your result in scientific notation in SI units.
(a) Translate the following information into symbols, using the
notation with two subscripts introduced in section 2.5. Eowyn is
riding on her horse at a velocity of 11 m/s. She twists around in
her saddle and fires an arrow backward. Her bow fires arrows at 25
m/s. (b) Find the speed of the arrow relative to the ground.
Our full discussion of two- and three-dimensional motion is
postponed until the second half of the book, but here is a chance to
use a little mathematical creativity in anticipation of that generalization.
Suppose a ship is sailing east at a certain speed v, and a
passenger is walking across the deck at the same speed v, so that
his track across the deck is perpendicular to the ship's center-line.
What is his speed relative to the water, and in what direction is he
moving relative to the water?
Solution, p. 276
Freddi Fish(TM) has a position as a function of time given by
x = a/(b + t2). Find her maximum speed.
Driving along in your car, you take your foot off the gas,
and your speedometer shows a reduction in speed. Describe a frame
of reference in which your car was speeding up during that same
period of time. (The frame of reference should be defined by an
observer who, although perhaps in motion relative to the earth, is
not changing her own speed or direction of motion.)
The figure shows the motion of a bluefin tuna, as measured
by a radio tag (Block et al., Nature, v. 434, p. 1121, 2005), over
the course of several years. Until this study, it had been believed
that the populations of the fish in the eastern and western Atlantic
were separate, but this particular fish was observed to cross the
entire Atlantic Ocean, from Virginia to Ireland. Points A, B, and C
show a period of one month, during which the fish made the most
rapid progress. Estimate its speed during that month, in units of
kilometers per hour.
Galileo's contradiction of Aristotle had serious consequences. He was
interrogated by the Church authorities and convicted of teaching that the
earth went around the sun as a matter of fact and not, as he had promised
previously, as a mere mathematical hypothesis. He was placed under permanent
house arrest, and forbidden to write about or teach his theories.
Immediately after being forced to recant his claim that the earth revolved
around the sun, the old man is said to have muttered defiantly "and yet
it does move." The story is dramatic, but there are some omissions in
the commonly taught heroic version. There was a rumor that the Simplicio
character represented the Pope. Also, some of the ideas Galileo
advocated had controversial religious overtones. He believed in the existence
of atoms, and atomism was thought by some people to contradict
the Church's doctrine of transubstantiation, which said that in the Catholic
mass, the blessing of the bread and wine literally transformed them into
the flesh and blood of Christ. His support for a cosmology in which the
earth circled the sun was also disreputable because one of its supporters,
Giordano Bruno, had also proposed a bizarre synthesis of Christianity
with the ancient Egyptian religion.