### Additional Problems

(a) At what relative velocity is $\gamma =1.50?$ (b) At what relative velocity is $\gamma =100?$

(a) At what relative velocity is $\gamma =2.00?$ (b) At what relative velocity is $\gamma =10.0?$

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

(a) How long does it take the astronaut in Example 5.5 to travel 4.30 ly at $0.99944c$ (as measured by the earthbound observer)? (b) How long does it take according to the astronaut? (c) Verify that these two times are related through time dilation with $\gamma =30.00$ as given.

(a) How fast would an athlete need to be running for a 100-$\text{m}$ race to look 100 yd long? (b) Is the answer consistent with the fact that relativistic effects are difficult to observe in ordinary circumstances? Explain.

(a) Find the value of $\gamma $ for the following situation. An astronaut measures the length of their spaceship to be 100 m, while an earthbound observer measures it to be 25.0 m. (b) What is the speed of the spaceship relative to Earth?

A clock in a spaceship runs one-tenth the rate at which an identical clock on Earth runs. What is the speed of the spaceship?

An astronaut has a heartbeat rate of 66 beats per minute as measured during his physical exam on Earth. The heartbeat rate of the astronaut is measured when he is in a spaceship traveling at 0.5*c* with respect to Earth by an observer (A) in the ship and by an observer (B) on Earth. (a) Describe an experimental method by which observer B on Earth will be able to determine the heartbeat rate of the astronaut when the astronaut is in the spaceship. (b) What will be the heartbeat rate(s) of the astronaut reported by observers A and B?

A spaceship (A) is moving at speed *c/*2 with respect to another spaceship (B). Observers in A and B set their clocks so that the event at (*x, y, z, t*) of turning on a laser in spaceship B has coordinates (0*,* 0*,* 0*,* 0) in A and also (0*,* 0*,* 0*,* 0) in B. An observer at the origin of B turns on the laser at $t=0$ and turns it off at $t=\tau $ in his time. What is the time duration between on and off as seen by an observer in A?

Same two observers as in the preceding exercise, but now we look at two events occurring in spaceship A. A photon arrives at the origin of A at its time $t=0$ and another photon arrives at $(x=1.00\phantom{\rule{0.2em}{0ex}}\text{m},0,0)$ at $t=0$ in the frame of ship A. (a) Find the coordinates and times of the two events as seen by an observer in frame B. (b) In which frame are the two events simultaneous and in which frame are they are not simultaneous?

Same two observers as in the preceding exercises. A rod of length 1 m is laid out on the *x*-axis in the frame of B from origin to $\left(x=1.00\phantom{\rule{0.2em}{0ex}}\text{m},0,0\right).$ What is the length of the rod observed by an observer in the frame of spaceship A?

An observer at origin of inertial frame S sees a flashbulb go off at $x=150\phantom{\rule{0.2em}{0ex}}\text{km},y=15.0\phantom{\rule{0.2em}{0ex}}\text{km},$ and $z=1.00\phantom{\rule{0.2em}{0ex}}\text{km}$ at time $t=4.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}\text{s}.$ At what time and position in the S$\prime $ system did the flash occur, if S$\prime $ is moving along shared *x*-direction with S at a velocity $v=0.6c?$

An observer sees two events $1.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-8}}\phantom{\rule{0.2em}{0ex}}\text{s}$ apart at a separation of 800 m. How fast must a second observer be moving relative to the first to see the two events occur simultaneously?

An observer standing by the railroad tracks sees two bolts of lightning strike the ends of a 500-m-long train simultaneously at the instant the middle of the train passes him at 50 m/s. Use the Lorentz transformation to find the time between the lightning strikes as measured by a passenger seated in the middle of the train.

Two astronomical events are observed from Earth to occur at a time of 1 s apart and a distance separation of $1.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{9}\text{m}$ from each other. (a) Determine whether separation of the two events is space like or time like. (b) State what this implies about whether it is consistent with special relativity for one event to have caused the other?

Two astronomical events are observed from Earth to occur at a time of 0.30 s apart and a distance separation of $2.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{9}\text{m}$ from each other. How fast must a spacecraft travel from the site of one event toward the other to make the events occur at the same time when measured in the frame of reference of the spacecraft?

A spacecraft starts from being at rest at the origin and accelerates at a constant rate *g*, as seen from Earth, taken to be an inertial frame, until it reaches a speed of *c/*2. (a) Show that the increment of proper time is related to the elapsed time in Earth’s frame by:

(b) Find an expression for the elapsed time to reach speed *c/*2 as seen in Earth’s frame. (c) Use the relationship in (a) to obtain a similar expression for the elapsed proper time to reach *c*/2 as seen in the spacecraft, and determine the ratio of the time seen from Earth with that on the spacecraft to reach the final speed.

(a) All but the closest galaxies are receding from our own Milky Way Galaxy. If a galaxy $12.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{9}\phantom{\rule{0.2em}{0ex}}\text{ly}$ away is receding from us at 0.900*c*, at what velocity relative to us must we send an exploratory probe to approach the other galaxy at 0.990*c* as measured from that galaxy? (b) How long will it take the probe to reach the other galaxy as measured from 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.)

Suppose a spaceship heading straight toward the Earth at 0.750*c* can shoot a canister at 0.500*c* relative to the ship. (a) What is the velocity of the canister relative to Earth, if it is shot directly at Earth? (b) If it is shot directly away from Earth?

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

(a) Suppose the speed of light were only 3000 m/s. A jet fighter moving toward a target on the ground at 800 m/s shoots bullets, each having a muzzle velocity of 1000 m/s. What are the bullets’ velocity relative to the target? (b) If the speed of light was this small, would you observe relativistic effects in everyday life? Discuss.

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

A space probe speeding towards the nearest star moves at $0.250c$ and sends radio information at a broadcast frequency of 1.00 GHz. What frequency is received on Earth?

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?

(a) Calculate the speed of a $1.00\text{-}\text{\mu g}$ particle of dust that has the same momentum as a proton moving at 0.999*c*. (b) What does the small speed tell us about the mass of a proton compared to even a tiny amount of macroscopic matter?

(a) Calculate $\gamma $ for a proton that has a momentum of $1.00\phantom{\rule{0.2em}{0ex}}\text{kg}\xb7\text{m/s}\text{.}$ (b) What is its speed? Such protons form a rare component of cosmic radiation with uncertain origins.

Show that the relativistic form of Newton’s second law is (a) $F=m\frac{du}{dt}\phantom{\rule{0.2em}{0ex}}\frac{1}{{\left(1-{u}^{2}\text{/}{c}^{2}\right)}^{3\text{/}2}};$ (b) Find the force needed to accelerate a mass of 1 kg by 1 ${\text{m/s}}^{2}$ when it is traveling at a velocity of *c*/2.

A positron is an antimatter version of the electron, having exactly the same mass. When a positron and an electron meet, they annihilate, converting all of their mass into energy. (a) Find the energy released, assuming negligible kinetic energy before the annihilation. (b) If this energy is given to a proton in the form of kinetic energy, what is its velocity? (c) If this energy is given to another electron in the form of kinetic energy, what is its velocity?

What is the kinetic energy in MeV of a π-meson that lives $1.40\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}1{0}^{\mathrm{-16}}\phantom{\rule{0.2em}{0ex}}\text{s}$ as measured in the laboratory, and $0.840\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}1{0}^{\mathrm{-16}}\phantom{\rule{0.2em}{0ex}}\text{s}$ when at rest relative to an observer, given that its rest energy is 135 MeV?

Find the kinetic energy in MeV of a neutron with a measured life span of 2065 s, given its rest energy is 939.6 MeV, and rest life span is 900s.

(a) Show that ${(pc)}^{2}\text{/}{(m{c}^{2})}^{2}={\gamma}^{2}-1.$ This means that at large velocities $pc>>m{c}^{2}.$ (b) Is $E\approx pc$ when $\gamma =30.0,$ as for the astronaut discussed in the twin paradox?

One cosmic ray neutron has a velocity of $0.250c$ relative to the Earth. (a) What is the neutron’s total energy in MeV? (b) Find its momentum. (c) Is $E\approx pc$ in this situation? Discuss in terms of the equation given in part (a) of the previous problem.

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

(a) What is the effective accelerating potential for electrons at the Stanford Linear Accelerator, if $\gamma =1.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}$ for them? (b) What is their total energy (nearly the same as kinetic in this case) in GeV?

(a) Using data from Potential Energy of a System, 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 $750{\text{kg/m}}^{3},$ what is the ratio of mass destroyed to original mass, $\text{\Delta}m\text{/}m?$

(a) Calculate the energy released by the destruction of 1.00 kg of mass. (b) How many kilograms could be lifted to a 10.0 km height by this amount of energy?

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?

Suppose you use an average of $500\phantom{\rule{0.2em}{0ex}}\text{kW}\xb7\text{h}$ of electric energy per month in your home. (a) How long would 1.00 g of mass converted to electric energy with an efficiency of 38.0% last you? (b) How many homes could be supplied at the $500\phantom{\rule{0.2em}{0ex}}\text{kW}\xb7\text{h}$ per month rate for one year by the energy from the described mass conversion?

(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 ${10}^{4}\phantom{\rule{0.2em}{0ex}}\text{kg}?$

Nuclear-powered rockets were researched for some years before safety concerns became paramount. (a) What fraction of a rocket’s mass would have to be destroyed to get it into a low Earth orbit, neglecting the decrease in gravity? (Assume an orbital altitude of 250 km, and calculate both the kinetic energy (classical) and the gravitational potential energy needed.) (b) If the ship has a mass of $1.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{kg}$ (100 tons), what total yield nuclear explosion in tons of TNT is needed?

The sun produces energy at a rate of $3.85\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{26}$ W by the fusion of hydrogen. About 0.7% of each kilogram of hydrogen goes into the energy generated by the Sun. (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)?

Show that ${E}^{2}-{p}^{2}{c}^{2}$ for a particle is invariant under Lorentz transformations.