Problems

(a) What is $\gamma$ if $v=0.250c?$ (b) If $v=0.500c?$

Particles called $\pi$-mesons are produced by accelerator beams. If these particles travel at $2.70\times {10}^8\text{m/s}$ and live $2.60\times {10}^{\mathrm{-8}}\text{s}$ when at rest relative to an observer, how long do they live as viewed in the laboratory?

A neutral $\pi$-meson is a particle that can be created by accelerator beams. If one such particle lives $1.40\times {10}^{\mathrm{-16}}\text{s}$ as measured in the laboratory, and $0.840\times {10}^{\mathrm{-16}}\text{s}$ when at rest relative to an observer, what is its velocity relative to the laboratory?

If relativistic effects are to be less than 1%, then $\gamma$ must be less than 1.01. At what relative velocity is $\gamma =1.01?$

A spaceship, 200 m long as seen on board, moves by the Earth at 0.970c. What is its length as measured by an earthbound observer?

(a) How far does the muon in Example 5.3 travel according to the earthbound observer? (b) How far does it travel as viewed by an observer moving with it? Base your calculation on its velocity relative to the Earth and the time it lives (proper time). (c) Verify that these two distances are related through length contraction $\gamma =3.20.$

Unreasonable Results A spaceship is heading directly toward Earth at a velocity of 0.800c. The astronaut on board claims that he can send a canister toward the Earth at 1.20c relative to Earth. (a) Calculate the velocity the canister must have relative to the spaceship. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Describe what happens to the angle $\alpha =\text{tan}\left(v\text{/}c\right),$ and therefore to the transformed axes in Figure 5.17, as the relative velocity v of the S and $\text{S}\prime$ frames of reference approaches c.

A man standing still at a train station watches two boys throwing a baseball in a moving train. Suppose the train is moving east with a constant speed of 20 m/s and one of the boys throws the ball with a speed of 5 m/s with respect to himself toward the other boy, who is 5 m west from him. What is the velocity of the ball as observed by the man on the station?

A man is running on a straight road perpendicular to a train track and away from the track at a speed of 12 m/s. The train is moving with a speed of 30 m/s with respect to the track. What is the speed of the man with respect to a passenger sitting at rest in the train?

In a frame at rest with respect to the billiard table, a billiard ball of mass m moving with speed v strikes another billiard ball of mass m at rest. The first ball comes to rest after the collision while the second ball takes off with speed v in the original direction of the motion of the first ball. This shows that momentum is conserved in this frame. (a) Now, describe the same collision from the perspective of a frame that is moving with speed v in the direction of the motion of the first ball. (b) Is the momentum conserved in this frame?

In a frame S, two events are observed: event 1: a pion is created at rest at the origin and event 2: the pion disintegrates after time $\tau$. Another observer in a frame $\text{S}\prime$ is moving in the positive direction along the positive x-axis with a constant speed v and observes the same two events in his frame. The origins of the two frames coincide at $t=t\prime =0.$ (a) Find the positions and timings of these two events in the frame $\text{S}\prime$ (a) according to the Galilean transformation, and (b) according to the Lorentz transformation.

Two planets are on a collision course, heading directly toward each other at 0.250c. A spaceship sent from one planet approaches the second at 0.750c as seen by the second planet. What is the velocity of the ship relative to the first planet?

What is the relative velocity of two spaceships if one fires a missile at the other at 0.750c and the other observes it to approach at 0.950c?

Show that for any relative velocity v between two observers, a beam of light projected by one directly away from the other will move away at the speed of light (provided that v is less than c, of course).

Find the momentum of a helium nucleus having a mass of $6.68\times {10}^{\mathrm{-27}}\text{kg}$ that is moving at 0.200c.

(a) Find the momentum of a $1.00\times {10}^9\text{-kg}$ asteroid heading towards Earth at 30.0 km/s. (b) Find the ratio of this momentum to the classical momentum. (Hint: Use the approximation that $\gamma =1+(1\text{/}2)v^2\text{/}{c}^2$ at low velocities.)

What is the velocity of an electron that has a momentum of $3.04\times {10}^{\mathrm{-21}}\text{kg}·\text{m/s}$ ? Note that you must calculate the velocity to at least four digits to see the difference from c.

What is the rest energy of an electron, given its mass is $9.11\times {10}^{-31}\text{kg}?$ Give your answer in joules and MeV.

If the rest energies of a proton and a neutron (the two constituents of nuclei) are 938.3 and 939.6 MeV, respectively, what is the difference in their mass in kilograms?

A supernova explosion of a $2.00\times {10}^{31}\text{kg}$ star produces $1.00\times {10}^{44}\text{J}$ of energy. (a) How many kilograms of mass are converted to energy in the explosion? (b) What is the ratio $\text{Δ}m\text{/}m$ of mass destroyed to the original mass of the star?

(a) Using data from this table, calculate the amount of mass converted to energy by the fusion of 1.00 kg of hydrogen. (b) What is the ratio of mass destroyed to the original mass, $\text{Δ}m\text{/}m$ ? (c) How does this compare with $\text{Δ}m\text{/}m$ for the fission of 1.00 kg of uranium?

A muon has a rest mass energy of 105.7 MeV, and it decays into an electron and a massless particle. (a) If all the lost mass is converted into the electron’s kinetic energy, find $\gamma$ for the electron. (b) What is the electron’s velocity?

(a) Calculate the relativistic kinetic energy of a 1000-kg car moving at 30.0 m/s if the speed of light were only 45.0 m/s. (b) Find the ratio of the relativistic kinetic energy to classical.

(a) Beta decay is nuclear decay in which an electron is emitted. If the electron is given 0.750 MeV of kinetic energy, what is its velocity? (b) Comment on how the high velocity is consistent with the kinetic energy as it compares to the rest mass energy of the electron.