Problem Set 9
Due: see website for due date
Chapter 28: Special Relativity
Questions: 2, 4, 6, 7
Problems: 4, 6, 12, 14, 21, 27, 32
Q28.2: On a highway there is a flashing light to mark the start of a section of the road
where work is being done. Who measures the proper time be- tween two flashes of
light? (a) A worker standing still on the road (b) A driver in a car approaching at a
constant velocity (c) Both the worker and the driver (d) Neither the worker nor the driver.
Q28.4: Two spacecrafts A and B are moving relative to each other at a con- stant
velocity. Observers in spacecraft A see spacecraft B. Likewise, observers in spacecraft
B see spacecraft A. Who sees the proper length of either spacecraft? (a) Observers in
spacecraft A see the proper length of spacecraft B. (b) Observers in spacecraft B see
the proper length of spacecraft A. (c) Observers in both spacecrafts see the proper
length of the other spacecraft. (d) Observers in neither spacecraft see the proper length
of the other spacecraft.
Q28.6: In a baseball game the batter hits the ball into center field and takes off for first
base. The catcher can only stand and watch. Assume that the batter runs at a constant
velocity. Who measures the proper time it takes for the runner to reach first base, and
who measures the proper length between home plate and first base? (a) The catcher
measures the proper time, and the runner measures the proper length. (b) The runner
measures the proper time, and the catcher measures the proper length. (c) The catcher
measures both the proper time and the proper length. (d) The runner measures both the
proper time and the proper length.
Q28.7: To which one or more of the following situations do the time- dilation and lengthcontraction equations apply? (a) With respect to an inertial frame, two observers have
different constant accelerations. (b) With respect to an inertial frame, two observers have
the same constant acceleration. (c) With respect to an inertial frame, two observers are
moving with different constant velocities. (d) With respect to an iner- tial frame, one
observer has a constant velocity, and another observer has a constant acceleration. (e)
All of the above.
P28.4: Suppose that you are traveling on board a spacecraft that is moving with respect
to the earth at a speed of 0.975c. You are breathing at a rate of 8.0 breaths per minute.
As monitored on earth, what is your breathing rate?
Answer: 1.8 breaths/minute
P28.6: A spaceship travels at a constant speed from earth to a planet or- biting another
star. When the spacecraft arrives, 12 years have elapsed on earth, and 9.2 years have
elapsed on board the ship. How far away (in meters) is the planet, according to
observers on earth? Answer: 7.3×1016 m.
P28.12: A Martian leaves Mars in a spaceship that is heading to Venus. On the way, the
spaceship passes earth with a speed v = 0.80c relative to it. Assume that the three
planets do not move relative to each other during the trip. The distance between Mars
and Venus is 1.20 × 1011 m, as measured by a person on earth. (a) What does the
Martian measure for the distance between Mars and Venus? (b) What is the time of the
trip (in seconds) as measured by the Martian? Answer: 7.2×1010 m, 300 s
1
P28.14: An unstable high-energy particle is created in the laboratory, and it moves at a
speed of 0.990c. Relative to a stationary reference frame fixed to the laboratory, the
particle travels a distance of 1.05×10-3 m before disintegrating. What are (a) the proper
distance and (b) the distance measured by a hypothetical person traveling with the
particle? Determine the particle’s (c) proper lifetime and (d) its dilated lifetime.
Answer: 1.05 mm, 14.8 mm, 4.98×10-13s, 3.53×10-12s
P28.21: A woman is 1.6 m tall and has a mass of 55 kg. She moves past an observer
with the direction of the motion parallel to her height. The observer measures her
relativistic momentum to have a magnitude of 2.0×1010 kg? m/s. What does the
observer measure for her height? Answer: 1.0 m
Hint: use momentum to determine v first.
P28.27: Suppose that one gallon of gasoline produces 1.1×108 J of energy, and this
energy is sufficient to operate a car for twenty miles. An aspirin tablet has a mass of 325
mg. If the aspirin could be converted completely into thermal energy, how many miles
could the car go on a single tablet? Answer: 5.3×106 mi
Solution
The mass m of the aspirin is related to its rest energy E0 by Equation 28.5, E0 = mc2. Since it
requires 1.1  108 J to operate the car for twenty miles, we can calculate the number of miles that
the car can go on the energy that is equivalent to the mass of one tablet. We begin by converting
the mass m from milligrams (mg) to kilograms (kg):

6
 325 mg   1000 mg 
 1000 g   325 10 kg
1g

1 kg


The number N of miles the car can go on one aspirin tablet is
N
mc
2
1.1  10 J  /  20.0 mi 
8
325  106 kg  3.0  108 m/s 


1.1  10 J  /  20.0 mi 
8
2
 5.3  106 mi
P28.32: An electron is accelerated from rest through a potential difference that has a
magnitude of 2.40 3 107 V. The mass of the electron is 9.11×10-31 kg, and the negative
charge of the electron has a magnitude of 1.60×10-19 C. (a) What is the relativistic kinetic
energy (in joules) of the electron? (b) What is the speed of the electron? Express your
answer as a multiple of c, the speed of light in a vacuum.
Answer: 3.84×10-12 J, 0.999781c
2