97
Problem 14.
x
v
t
t
Homework Problems
13
.
. 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–ct+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 make
sense.
(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(e
x
)/dx=e
x
.]
(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.
14 S. 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.
15 S. 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.
16 S. 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.
17 S
.
. A honeybee’s position as a function of time is given by x=10t-t
3
,
where t is in seconds and x in meters. What is its acceleration at t=3.0 s.
18 S. 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 accelera-
tion 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.