Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information....

Homework Problems

Astronauts in three different spaceships are communicating with each other. Those aboard ships A and B agree on the rate at which time is passing, but they disagree with the ones on ship C.

(a) Describe the motion of the other two ships according to Alice, who is aboard ship A.

(b) Give the description according to Betty, whose frame of reference is ship B.

(c) Do the same for Cathy, aboard ship C.


(a) Figure g on page 19 is based on a light clock moving at a certain speed, v. By measuring with a ruler on the figure, determine v/c.

(b) By similar measurements, find the time contraction factor γ, which equals T/t.

(c) Locate your numbers from parts a and b as a point on the graph in figure h on page 20, and check that it actually lies on the curve. Make a sketch showing where the point is on the curve.


This problem is a continuation of problem 2. Now imagine that the spaceship speeds up to twice the velocity. Draw a new triangle on the same scale, using a ruler to make the lengths of the sides accurate. Repeat parts b and c for this new diagram.
What happens in the equation for γ when you put in a negative number for v? Explain what this means physically, and why it makes sense.
(a) By measuring with a ruler on the graph in figure m on page 24, estimate the γ values of the two supernovae.

(b) Figure m gives the values of v/c. From these, compute γ values and compare with the results from part a.

(c) Locate these two points on the graph in figure h, and make a sketch showing where they lie.


The Voyager 1 space probe, launched in 1977, is moving faster relative to the earth than any other human-made object, at 17,000 meters per second.

(a) Calculate the probe's γ

(b) Over the course of one year on earth, slightly less than one year passes on the probe. How much less? (There are 31 million seconds in a year.) p


(a) A free neutron (as opposed to a neutron bound into an atomic nucleus) is unstable, and undergoes beta decay (which you may want to review). The masses of the particles involved are as follows:

neutron 1.67495 10-27 kg proton 1.67265 10-27 kg electron 0.00091 10-27 kg antineutrino < 10-35 kg

Find the energy released in the decay of a free neutron.

(b) Neutrons and protons make up essentially all of the mass of the ordinary matter around us. We observe that the universe around us has no free neutrons, but lots of free protons (the nuclei of hydrogen, which is the element that 90% of the universe is made of). We find neutrons only inside nuclei along with other neutrons and protons, not on their own.

If there are processes that can convert neutrons into protons, we might imagine that there could also be proton-to-neutron conversions, and indeed such a process does occur sometimes in nuclei that contain both neutrons and protons: a proton can decay into a neutron, a positron, and a neutrino. A positron is a particle with the same properties as an electron, except that its electrical charge is positive (see chapter 7). A neutrino, like an antineutrino, has negligible mass.

Although such a process can occur within a nucleus, explain why it cannot happen to a free proton. (If it could, hydrogen would be radioactive, and you wouldn't exist!)


(a) Find a relativistic equation for the velocity of an object in terms of its mass and momentum (eliminating γ).

(b) Show that your result is approximately the same as the classical value, p/m, at low velocities.

(c) Show that very large momenta result in speeds close to the speed of light.


a)√ c)*
(a) Show that for v = (3/5)c, γ comes out to be a simple fraction.

(b) Find another value of v for which γ is a simple fraction.


10 An object moving at a speed very close to the speed of light is referred to as ultrarelativistic. Ordinarily (luckily) the only ultrarelativistic objects in our universe are subatomic particles, such as cosmic rays or particles that have been accelerated in a particle accelerator.

(a) What kind of number is γ for an ultrarelativistic particle?

(b) Repeat example 5 on page 35, but instead of very low, nonrelativistic speeds, consider ultrarelativistic speeds.

(c) Find an equation for the ratio E/p. The speed may be relativistic, but don't assume that it's ultrarelativistic.

(d) Simplify your answer to part c for the case where the speed is ultrarelativistic.

(e) We can think of a beam of light as an ultrarelativistic object - it certainly moves at a speed that's sufficiently close to the speed of light! Suppose you turn on a one-watt flashlight, leave it on for one second, and then turn it off. Compute the momentum of the recoiling flashlight, in units of kgm/s.

(f) Discuss how your answer in part e relates to the correspondence principle.


11 As discussed in book 3 of this series, the speed at which a disturbance travels along a string under tension is given by

where μ is the mass per unit length, and T is the tension.

(a) Suppose a string has a density ρ, and a cross-sectional area A. Find an expression for the maximum tension that could possibly exist in the string without producing v > c, which is impossible according to relativity. Express your answer in terms of ρ, A, and c. The interpretation is that relativity puts a limit on how strong any material can be.

(b) Every substance has a tensile strength, defined as the force per unit area required to break it by pulling it apart. The tensile strength is measured in units of N/m2, which is the same as the pascal (Pa), the mks unit of pressure. Make a numerical estimate of the maximum tensile strength allowed by relativity in the case where the rope is made out of ordinary matter, with a density on the same order of magnitude as that of water. (For comparison, kevlar has a tensile strength of about 4 109 Pa, and there is speculation that fibers made from carbon nanotubes could have values as high as 6 1010 Pa.)

(c) A black hole is a star that has collapsed and become very dense, so that its gravity is too strong for anything ever to escape from it. For instance, the escape velocity from a black hole is greater than c, so a projectile can't be shot out of it. Many people, when they hear this description of a black hole in terms of an escape velocity greater than c, wonder why it still wouldn't be possible to extract an object from a black hole by other means than launching it out as a projectile. For example, suppose we lower an astronaut into a black hole on a rope, and then pull him back out again. Why might this not work?


12 The earth is orbiting the sun, and therefore is contracted relativistically in the direction of its motion. Compute the amount by which its diameter shrinks in this direction.




Last Update: 2010-11-11