The control panel on a spaceship contains a light that blinks every 2.00 s as observed by an astronaut in the ship. If the spaceship is moving past Earth with a speed of $0.750 c,$ determine (a) the proper time interval between blinks and (b) the time interval between blinks as observed by a person on Earth.

Zulfiqar A.

Numerade Educator

A spaceship moves past Earth with a speed of 0.900$c .$ As it is passing, a person on Earth measures the spaceship's length to be 75.0 m . (a) Determine the spaceship's proper length. (b) Determine the time required for the spaceship to pass a point on Earth as measured by a person on Earth and (c) by an astronaut on board the spaceship.

Zulfiqar A.

Numerade Educator

If astronauts could travel at $v=0.950 c,$ we on Earth would say it takes $(4.20 / 0.950)=4.42$ years to reach Alpha Centauri, 4.20 light-years away. The astronauts disagree. (a) How much time passes on the astronauts' clocks? (b) What is the distance to Alpha Centauri as measured by the astronauts?

Zulfiqar A.

Numerade Educator

A meter stick moving at 0.900$c$ relative to the Earth's surface approaches an observer at rest with respect to the Earth’s surface. (a) What is the meter stick’s length as measured by the observer? (b) Qualitatively, how would the answer to part (a) change if the observer started running toward the meter stick?

Zulfiqar A.

Numerade Educator

The length of a moving spaceship is 28.0 m according to an astronaut on the spaceship. If the spaceship is contracted by 15.0 cm according to an Earth observer, what is the speed of the spaceship?

Zulfiqar A.

Numerade Educator

An astronaut at rest on Earth has a heart rate of 70. beats/min. When the astronaut is traveling in a spaceship at 0.90$c,$ what will this rate be as measured by (a) an observer also in the ship and (b) an observer at rest on Earth?

Zulfiqar A.

Numerade Educator

The average lifetime of a pi meson in its own frame of reference (i.e., the proper lifetime) is $2.6 \times 10^{-8} s.$ If the meson moves with a speed of 0.98$c,$ what is (a) its mean lifetime as measured by an observer on Earth, and (b) the average distance it travels before decaying, as measured by an observer on Earth? (c) What distance would it travel if time dilation did not occur?

Zulfiqar A.

Numerade Educator

An astronaut is traveling in a space vehicle that has a speed of 0.500$c$ relative to Earth. The astronaut measures his pulse rate at 75.0 beats per minute. Signals generated by the astronaut’s pulse are radioed to Earth when the vehicle is moving perpendicular to a line that connects the vehicle with an Earth observer. (a) What pulse rate does the Earth observer measure? (b) What would be the pulse rate if the speed of the space vehicle were increased to 0.990$c$?

Zulfiqar A.

Numerade Educator

A muon formed high in Earth’s atmosphere travels toward Earth at a speed $v=0.990 c$ for a distance of 4.60 km as measured by an observer at rest with respect to Earth. It then decays into an electron, a neutrino, and an antineutrino. (a) How long does the muon survive according to an observer at rest on Earth? (b) Compute the gamma factor associated with the muon. (c) How much time passes according to an observer traveling with the muon? (d) What distance does the muon travel according to an observer traveling with the muon? (e) A third observer traveling toward the muon at $c/2$ measures the lifetime of the particle. According to this observer, is the muon’s lifetime shorter or longer than the lifetime measured by the observer at rest with respect to Earth? Explain.

Zulfiqar A.

Numerade Educator

A star is 15.0 light - years (ly) from Earth. (a) At what constant speed must a spacecraft travel on its journey to the star so that the Earth–star distance measured by an astronaut on board the spacecraft is 3.00 ly? (b) What is the journey’s travel time in years as measured by a person on Earth and (c) by the astronaut?

Zulfiqar A.

Numerade Educator

The proper length of one spaceship is three times that of another. The two spaceships are traveling in the same direction and, while both are passing overhead, an Earth observer measures the two spaceships to have the same length. If the slower spaceship has a speed of 0.350$c$ with respect to Earth, determine the speed of the faster spaceship.

Zulfiqar A.

Numerade Educator

A car traveling at 35.0 m/s takes 26.0 minutes to travel a certain distance according to the driver’s clock in the car. How long does the trip take according to an observer at rest on Earth? Hint: The following approximation is helpful: $[1-x]^{-\frac{1}{2}} \approx 1+\frac{1}{2} x$ for $x<<1$

Zulfiqar A.

Numerade Educator

A super train of proper length $1.00 \times 10^{2} \mathrm{m}$ travels at a speed of 0.95$c$ as it passes through a tunnel having proper length 50.0 m. As seen by a track side observer, is the train ever completely within the tunnel? If so, by how much?

Zulfiqar A.

Numerade Educator

A box is cubical with sides of proper lengths $L_{1}=L_{2}=L_{3},$ as shown in Figure P 26.14, when viewed in its own rest frame. If this block moves parallel to one of its edges with a speed of 0.80$c$ past an observer, (a) what shape does it appear to have to this observer? (b) What is the length of each side as measured by the observer?

Zulfiqar A.

Numerade Educator

(a) What is the momentum of a proton moving at 0.900$c$?

(b) At what speed will a particle’s relativistic momentum equal twice its classical momentum?

Zulfiqar A.

Numerade Educator

At what speed do the classical and relativistic values of a particle’s momentum differ by 10.0%?

Zulfiqar A.

Numerade Educator

An electron has a momentum with magnitude three times the magnitude of its classical momentum. (a) Find the speed of the electron. (b) How would your result change if the particle were a proton?

Zulfiqar A.

Numerade Educator

Calculate the classical momentum of a proton traveling at 0.990$c,$ neglecting relativistic effects. (b) Repeat the calculation while including relativistic effects. (c) Does it make sense to neglect relativity at such speeds ?

Zulfiqar A.

Numerade Educator

An unstable particle at rest breaks up into two fragments of unequal mass. The mass of the lighter fragment is equal to 2.50 $\times 10^{-28} \mathrm{kg}$ and that of the heavier fragment is $1.67 \times 10^{-27} \mathrm{kg}$ . If the lighter fragment has a speed of 0.893 $c$ after the breakup, what is the speed of the heavier fragment?

Zulfiqar A.

Numerade Educator

Spaceship $R$ is moving to the right at a speed of 0.70$c$ with respect to Earth. A second spaceship, $L,$ moves to the left at the same speed with respect to Earth. What is the speed of $L$ with respect to $R$ ?

Zulfiqar A.

Numerade Educator

An electron moves to the right with a speed of 0.90 $c$ relative to the laboratory frame. A proton moves to the left with a speed of 0.70 $c$ relative to the electron. Find the speed of the proton relative to the laboratory frame.

Zulfiqar A.

Numerade Educator

A spaceship travels at 0.750$c$ relative to Earth. If the spaceship fires a small rocket in the forward direction, how fast (relative to the ship) must it be fired for it to travel at 0.950$c$ relative to Earth?

Zulfiqar A.

Numerade Educator

A spaceship is moving away from Earth at 0.900$c$ when it fires a small rocket in the forward direction at 0.500$c$ relative to the spaceship. Calculate the rocket's speed relative to Earth.

Zulfiqar A.

Numerade Educator

Two identical spaceships with proper lengths of 175 m are launched from Earth. Spaceship A is launched in one direction at 0.500$c$ and spaceship $B$ is launched in the opposite direction at 0.750$c .$ (a) What is the speed of spaceship $B$ relative to spaceship $A ?$ (b) What is the length of spaceship $A$ as measured by astronauts on spaceship $B?$

Zulfiqar A.

Numerade Educator

Spaceship $A$ moves away from Earth at a speed of 0.800$c$ (Fig. P 26.25 ). Spaceship $B$ pursues at a speed of 0.900 c relative to Earth. Observers on Earth see $B$ overtaking $A$ at a relative speed of 0.100$c .$ With what speed is $B$ overtaking $A$ as seen by the crew of spaceship $B$ ?

Zulfiqar A.

Numerade Educator

A pulsar is a stellar object that emits light in short bursts. Suppose a pulsar with a speed of 0.950 $c$ approaches Earth, and a rocket with a speed of 0.995 $c$ heads toward the pulsar. (Both speeds are measured in Earth’s frame of reference.) If the pulsar emits 10.0 pulses per second in its own frame of reference, at what rate are the pulses emitted in the rocket’s frame of reference?

Zulfiqar A.

Numerade Educator

A rocket moves with a velocity of 0.92 $c$ to the right with respect to a stationary observer $A$ . An observer $B$ moving relative to observer $A$ finds that the rocket is moving with a velocity of 0.95$c$ to the left. What is the velocity of observer $B$ relative to observer $A$ ? (Hint: Consider observer $B^{\prime}s$ velocity in the frame of reference of the rocket.)

Zulfiqar A.

Numerade Educator

A proton moves with a speed of 0.950 $c.$ Calculate (a) its rest energy, (b) its total energy, and (c) its kinetic energy.

Zulfiqar A.

Numerade Educator

Protons in an accelerator at the Fermi National Laboratory near Chicago are accelerated to a total energy that is 400 times their rest energy. (a) What is the speed of these protons in terms of $c$ ? (b) What is their kinetic energy in MeV?

Zulfiqar A.

Numerade Educator

A proton in a large accelerator has a kinetic energy of 175 GeV. (a) Compare this kinetic energy to the rest energy of the proton, and find an approximate expression for the proton’s kinetic energy. (b) Find the speed of the proton.

Zulfiqar A.

Numerade Educator

What speed must a particle attain before its kinetic energy is double the value predicted by the non relativistic expression $K E=\frac{1}{2} m v^{2} ?$

Zulfiqar A.

Numerade Educator

Determine the energy required to accelerate an electron from (a) 0.500$c$ to 0.900$c$ and (b) 0.900$c$ to 0.990$c .$

Zulfiqar A.

Numerade Educator

A chain of nuclear reactions in the Sun’s core converts four protons into a helium nucleus. (a) What is the mass difference between four protons and a helium nucleus? (b) How much energy in MeV is released during the conversion of four protons into a helium nucleus?

Zulfiqar A.

Numerade Educator

An unstable particle with a mass equal to $3.34 \times 10^{-27} \mathrm{kg}$ is initially at rest. The particle decays into two fragments that fly off with velocities of 0.987$c$ and -0.868$c,$ respectively. Find the masses of the fragments. Hint: Conserve both mass-energy and momentum.

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Starting with the definitions of relativistic energy and momentum, show that $E^{2}=p^{2} c^{2}+m^{2} c^{4}$ (Eq. 26.13).

Zulfiqar A.

Numerade Educator

Consider the reaction $_{92}^{285} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow_{57}^{148} \mathrm{La}+_{35}^{87} \mathrm{Br}+_{0}^{1} \mathrm{n}.$ (a) Write the conservation of relativistic energy equation symbolically in terms of the rest energy and the kinetic energy, setting the initial total energy equal to the final total energy. (b) Using values from Appendix B, find the total mass of the initial particles. (c) Using the values given below, find the total mass of the particles after the reaction takes place. (d) Subtract the final particle mass from the initial particle mass. (e) Convert the answer to part (d) to MeV, obtaining the kinetic energy of the daughter particles. Neglect the kinetic energy of the reactants. Note: Lanthanum-148 has atomic mass 147.932 236 u; bromine - 87 has atomic mass 86.920 711 19 u.

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Consider electrons accelerated to a total energy of 20.0 GeV in the 3.00 - km - long Stanford Linear Accelerator. (a) What is the $\gamma$ factor for the electrons? (b) How long does the accelerator appear to the electrons? Electron mass energy: 0.511 MeV.

Zulfiqar A.

Numerade Educator

An electron has a speed of 0.750$c.$ (a) Find the speed of a proton that has the same kinetic energy as the electron. (b) Find the speed of a proton that has the same momentum as the electron.

Zulfiqar A.

Numerade Educator

The rest energy of an electron is 0.511 MeV. The rest energy of a proton is 938 MeV. Assume both particles have kinetic energies of 2.00 MeV. Find the speed of (a) the electron and (b) the proton. (c) By how much does the speed of the electron exceed that of the proton? Note: Perform the calculations in MeV; don’t convert the energies to joules. The answer is sensitive to rounding.

Zulfiqar A.

Numerade Educator

A spring of force constant $k$ is compressed by a distance $x$ from its equilibrium length. (a) Does the mass of the spring change when the spring is compressed? Explain. (b) Find an expression for the change in mass of the spring in terms of $k$ ,$x,$ and $c.$ (c) What is the change in mass if the force constant is $2.0 \times 10^{2} \mathrm{N} / \mathrm{m}$ and $x=15 \mathrm{cm} ?$

Zulfiqar A.

Numerade Educator

A star is 5.00 ly from the Earth. At what speed must a spacecraft travel on its journey to the star such that the Earth–star distance measured in the frame of the spacecraft is 2.00 ly?

Zulfiqar A.

Numerade Educator

An electron has a total energy equal to five times its rest energy. (a) What is its momentum? (b) Repeat for a proton.

Zulfiqar A.

Numerade Educator

An astronaut wishes to visit the Andromeda galaxy, making a one - way trip that will take 30.0 years in the spaceship’s frame of reference. Assume the galaxy is 2.00 million light - years away and his speed is constant. (a) How fast must he travel relative to Earth? (b) What will be the kinetic energy of his spacecraft, which has mass of $1.00 \times 10^{6} \mathrm{kg}?$ (c) What is the cost of this energy if it is purchased at a typical consumer price for electric energy, 13.0 cents per kWh? The following approximation will prove useful:

$$\frac{1}{\sqrt{1+x}} \approx 1-\frac{x}{2} \quad \text { for } x<<1$$

Zulfiqar A.

Numerade Educator

An alarm clock is set to sound in 10.0 h. At $t=0$ , the clock is placed in a spaceship moving with a speed of 0.75 $c$ (relative to Earth). What distance, as determined by an Earth observer, does the spaceship travel before the alarm clock sounds?

Zulfiqar A.

Numerade Educator

Owen and Dina are at rest in frame $\mathrm{S}^{\prime}$ , which is moving with a speed of 0.600$c$ with respect to frame $\mathrm{S}$ . They play a game of catch while Ed, at rest in frame $\mathrm{S}$ , watches the action (Fig. P26.45) . Owen throws the ball to Dina with a speed of 0.800 $c$ (according to Owen) and their separation (measured in $\mathrm{S}^{\prime} )$ is equal to $1.80 \times 10^{12} \mathrm{m} .$ (a) According to Dina, how fast is the ball moving? (b) According to Dina, what time interval is required for the ball to reach her? According to Ed, (c) how far apart are Owen and Dina, and (d) how fast is the ball moving?

Zulfiqar A.

Numerade Educator

An observer in a coasting spacecraft moves toward a mirror at speed $v$ relative to the reference frame labeled by $\mathrm{S}$ in Figure P 26.46. The mirror is stationary with respect to $\mathrm{S}$ . A light pulse emitted by the spacecraft travels toward the mirror and is reflected back to the spacecraft. The spacecraft is a distance $d$ from the mirror (as measured by observers in $\mathrm{S} )$ at the moment the light pulse leaves the spacecraft. What is the total travel time of the pulse as measured by observers in (a) the S frame and (b) the spacecraft?

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A spaceship of proper length 300. m takes 0.75 $\mu s$ to pass an Earth observer. Determine the speed of this spaceship as measured by the Earth observer.

Zulfiqar A.

Numerade Educator

The cosmic rays of highest energy are protons that have kinetic energy on the order of $10^{13}$ MeV. (a) From the point of view of the proton, how many kilometers across is the galaxy? (b) How long would it take a proton of this energy to travel across the Milky Way galaxy, having a diameter $\sim 10^{5}$ light-years, as measured in the proton's frame?

Zulfiqar A.

Numerade Educator

The non relativistic expression for the momentum of a particle, $p=m v,$ can be used if $v < < c .$ For what speed does the use of this formula give an error in the momentum of (a) 1.00$\%$ and (b) 10.0$\%$ ?

Zulfiqar A.

Numerade Educator

(a) Show that a potential difference of $1.02 \times 10^{6} \mathrm{V}$ would be sufficient to give an electron a speed equal to twice the speed of light if Newtonian mechanics remained valid at high speeds. (b) What speed would an electron actually acquire in falling through a potential difference equal to $1.02 \times 10^{6} \mathrm{V} ?$

Zulfiqar A.

Numerade Educator

The muon is an unstable particle that spontaneously decays into an electron and two neutrinos. In a reference frame in which the muons are stationary, if the number of muons at $t=0$ is $N_{0},$ the number at time $t$ is given by $N=N_{0} e^{-l / \tau},$ where $\tau$ is the mean lifetime, equal to 2.2$\mu$ s. Suppose the muons move at a speed of 0.95$c$ and there are $5.0 \times 10^{4}$ muons at $t=0 .$ (a) What is the observed lifetime of the muons? (b) How many muons remain after traveling a distance of 3.0 km?

Zulfiqar A.

Numerade Educator

Imagine that the entire Sun collapses to a sphere of radius $R_{g}$ such that the work required to remove a small mass $m$ from the surface would be equal to its rest energy $m c^{2} .$ This radius is called the gravitational radius for the Sun. Find $R_{g}$ . (It is believed that the ultimate fate of very massive stars is to collapse beyond their gravitational radii into black holes.)

Zulfiqar A.

Numerade Educator

The identical twins Speedo and Goslo join a migration from Earth to Planet $\mathrm{X}$ , which is 20.0 light-years away in a reference frame in which both planets are at rest. The twins, of the same age, depart at the same time on different spacecraft. Speedo's craft travels steadily at 0.950 $c,$ Goslo's at 0.750 $c.$ Calculate the age difference between the twins after Goslo's spacecraft lands on Planet $\mathrm{X}$ . Which twin is the older?

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An interstellar space probe is launched from Earth. After a brief period of acceleration, it moves with a constant velocity, 70.0% of the speed of light. Its nuclear - powered batteries supply the energy to keep its data transmitter active continuously. The batteries have a lifetime of 15.0 years as measured in a rest frame. (a) How long do the batteries on the space probe last as measured by mission control on Earth? (b) How far is the probe from Earth when its batteries fail as measured by mission control? (c) How far is the probe from Earth as measured by its built - in trip odometer when its batteries fail? (d) For what total time after launch are data received from the probe by mission control? Note that radio waves travel at the speed of light and fill the space between the probe and Earth at the time the battery fails.

Zulfiqar A.

Numerade Educator

An observer moving at a speed of 0.995 $c$ relative to a rod (Fig.P 26.55) measures its length to be 2.00 m and sees its length to be oriented at 30.0° with respect to its direction of motion. (a) What is the proper length of the rod? (b) What is the orientation angle in a reference frame moving with the rod?

Zulfiqar A.

Numerade Educator

An alien spaceship traveling 0.600$c$ toward Earth launches a landing craft with an advance guard of purchasing agents. The lander travels in the same direction with a velocity 0.800$c$ relative to the spaceship. As observed on Earth, the spaceship is 0.200 light-years from Earth when the lander is launched. (a) With what velocity is the lander observed to be approaching by observers on Earth? (b) What is the distance to Earth at the time of lander launch, as observed by the aliens on the mother ship? (c) How long does it take the lander to reach Earth as observed by the aliens on the mother ship? (d) If the lander has a mass of $4.00 \times 10^{5} \mathrm{kg}$ , what is its kinetic energy as observed in Earth's reference frame?

Zulfiqar A.

Numerade Educator