Problems & Exercises

(a) What is $\gamma$ if $v=0\text{.}\text{250}c$? (b) If $v=0\text{.}\text{500}c$?

Particles called $\pi$-mesons are produced by accelerator beams. If these particles travel at $2\text{.}\text{70}\times {\text{10}}^8\text{m/s}$ and live $2\text{.}\text{60}\times {\text{10}}^{-8}\text{s}$ 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 $1\text{.}\text{40}\times {\text{10}}^{-\text{16}}\text{s}$ as measured in the laboratory, and $0\text{.}\text{840}\times {\text{10}}^{-\text{16}}\text{s}$ 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 $\gamma =1\text{.}\text{01}$?

(a) At what relative velocity is $\gamma =1\text{.}\text{50}$? (b) At what relative velocity is $\gamma =\text{100}$?

Unreasonable Results

(a) Find the value of $\gamma$ for the following situation. An Earth-bound observer measures 23.9 h to have passed while signals from a high-velocity space probe indicate that $\text{24.0 h}$ have passed on board. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

A spaceship, 200 m long as seen on board, moves by the Earth at $0\text{.}\text{970}c$. What is its length as measured by an Earth-bound observer?

(a) How far does the muon in Example 28.1 travel according to the Earth-bound 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 $\text{γ=}3\text{.}\text{20}$.

(a) How long does it take the astronaut in Example 28.2 to travel 4.30 ly at $0\text{.99944}c$ (as measured by the Earth-bound observer)? (b) How long does it take according to the astronaut? (c) Verify that these two times are related through time dilation with $\text{γ=}\text{30}\text{.}\text{00}$ as given.

Unreasonable Results

(a) Find the value of $\gamma$ for the following situation. An astronaut measures the length of her spaceship to be 25.0 m, while an Earth-bound observer measures it to be 100 m. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Suppose a spaceship heading straight towards the Earth at $0\text{.}\text{750}c$ can shoot a canister at $0\text{.}\text{500}c$ relative to the ship. (a) What is the velocity of the canister relative to the Earth, if it is shot directly at the Earth? (b) If it is shot directly away from the Earth?

If a spaceship is approaching the Earth at $0.100c$ and a message capsule is sent toward it at $0.100c$ relative to the Earth, what is the speed of the capsule relative to the ship?

If a galaxy moving away from the Earth has a speed of $\mathrm{1000\; km/s}$ and emits $\text{656 nm}$ light characteristic of hydrogen (the most common element in the universe). (a) What wavelength would we observe on the Earth? (b) What type of electromagnetic radiation is this? (c) Why is the speed of the Earth in its orbit negligible here?

If two spaceships are heading directly towards each other at $0\text{.}\text{800}c$, at what speed must a canister be shot from the first ship to approach the other at $0\text{.}\text{999}c$ as seen by the second ship?

When a missile is shot from one spaceship towards another, it leaves the first at $0\text{.}\text{950}c$ and approaches the other at $0\text{.}\text{750}c$. What is the relative velocity of the two ships?

Near the center of our galaxy, hydrogen gas is moving directly away from us in its orbit about a black hole. We receive 1900 nm electromagnetic radiation and know that it was 1875 nm when emitted by the hydrogen gas. What is the speed of the gas?

Prove that for any relative velocity $v$ between two observers, a beam of light sent from one to the other will approach at speed $c$ (provided that $v$ is less than $c$, of course).

(a) All but the closest galaxies are receding from our own Milky Way Galaxy. If a galaxy $\text{12}\text{.}0\times {\text{10}}^9\text{ly}$ ly away is receding from us at 0.$0.900c$, at what velocity relative to us must we send an exploratory probe to approach the other galaxy at $0.990c$, as measured from that galaxy? (b) How long will it take the probe to reach the other galaxy as measured from the Earth? You may assume that the velocity of the other galaxy remains constant. (c) How long will it then take for a radio signal to be beamed back? (All of this is possible in principle, but not practical.)

Find the momentum of a helium nucleus having a mass of $6\text{.}\text{68}\times {\text{10}}^{\text{–27}}\text{kg}$ that is moving at $0\text{.}\text{200}c$.

(a) Find the momentum of a $1\text{.}\text{00}\times {\text{10}}^9\text{kg}$ asteroid heading towards the Earth at $\text{30.0 km/s}$. (b) Find the ratio of this momentum to the classical momentum. (Hint: Use the approximation that $\gamma =1+(1/2)v^2/c^2$ at low velocities.)

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

(a) Calculate the speed of a $\mathrm{1.00-}\mu \text{g}$ particle of dust that has the same momentum as a proton moving at $0\text{.}\text{999}c$. (b) What does the small speed tell us about the mass of a proton compared to even a tiny amount of macroscopic matter?

What is the rest energy of an electron, given its mass is $9\text{.}\text{11}\times {\text{10}}^{-\text{31}}\text{kg}$? 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 masses in kilograms?

A supernova explosion of a $2\text{.}\text{00}\times {\text{10}}^{\text{31}}\text{kg}$ star produces $1\text{.}\text{00}\times {\text{10}}^{\text{44}}\text{J}$ of energy. (a) How many kilograms of mass are converted to energy in the explosion? (b) What is the ratio $\mathrm{\Delta }m/m$ of mass destroyed to the original mass of the star?

(a) Using data from Table 7.1, 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, $\mathrm{\Delta }m/m$? (c) How does this compare with $\mathrm{\Delta }m/m$ 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.

What is the kinetic energy in MeV of a $\pi$-meson that lives $1\text{.}\text{40}\times {10}^{-\text{16}}\text{s}$ as measured in the laboratory, and $0\text{.}\text{840}\times {10}^{-\text{16}}\text{s}$ when at rest relative to an observer, given that its rest energy is 135 MeV?

(a) Show that $(\mathrm{pc}{)}^2/({\mathrm{mc}}^2{)}^2={\gamma }^2-1$. This means that at large velocities $\mathrm{pc}\text{>>}{\mathrm{mc}}^2$. (b) Is $E\approx \text{pc}$ when $\gamma =\text{30}\text{.}0$, as for the astronaut discussed in the twin paradox?

What is $\gamma$ for a proton having a mass energy of 938.3 MeV accelerated through an effective potential of 1.0 TV (teravolt) at Fermilab outside Chicago?

(a) Using data from Table 7.1, find the mass destroyed when the energy in a barrel of crude oil is released. (b) Given these barrels contain 200 liters and assuming the density of crude oil is $\mathrm{750\; kg}{\text{/m}}^3$, what is the ratio of mass destroyed to original mass, $\mathrm{\Delta }m/m$?

A Van de Graaff accelerator utilizes a 50.0 MV potential difference to accelerate charged particles such as protons. (a) What is the velocity of a proton accelerated by such a potential? (b) An electron?

(a) A nuclear power plant converts energy from nuclear fission into electricity with an efficiency of 35.0%. How much mass is destroyed in one year to produce a continuous 1000 MW of electric power? (b) Do you think it would be possible to observe this mass loss if the total mass of the fuel is ${10}^4\text{kg}$?

The Sun produces energy at a rate of $4\text{.}\text{00}\times {\text{10}}^{\text{26}}$ W by the fusion of hydrogen. (a) How many kilograms of hydrogen undergo fusion each second? (b) If the Sun is 90.0% hydrogen and half of this can undergo fusion before the Sun changes character, how long could it produce energy at its current rate? (c) How many kilograms of mass is the Sun losing per second? (d) What fraction of its mass will it have lost in the time found in part (b)?

Critical Thinking A space rock with a length of 1,000.0 m is moving through space at exactly 0.6 c. (a) If the space rock is moving toward an observer, what is the contracted length observed? (b) If the space rock is moving away from the observer, what is the contracted length observed? (c) Can the object reach the speed of light? (d) If the object were to stop in the observer’s reference frame, would it be observed to have proper length?