Relativity

An object of mass $m$ is initially at rest. after this object is accelerated to a speed of $2.40\times 10^{8}\frac{m}{s}$, it collides with and sticks to a second object of mass $m$ at rest. Immediately after the collision, what is the common speed of the two masses?

(A) $2.25\times 10^{8}\frac{m}{s}$
(B) $1.80\times 10^{8}\frac{m}{s}$
(C) $1.66\times 10^{8}\frac{m}{s}$
(D) $1.50\times 10^{8}\frac{m}{s}$
(E) $1.20\times 10^{8}\frac{m}{s}$

A person sets a one-meter long stick so that it makes a $30^{\circ}$ angle with the x-axis. An observer in a space ship moving along the x-axis measures the angle of the stick to be $60^{\circ}$ with the x-axis. With what speed is the spaceship moving in terms of the speed of light, $c$?

(A) $\displaystyle \frac{1}{2}c$
(B) $\displaystyle \frac{\sqrt{2}}{3}c$
(C) $\displaystyle \frac{3}{4}c$
(D) $\displaystyle \frac{2}{\sqrt{5}}c$
(E) $\displaystyle \frac{2\sqrt{2}}{3}c$

Two electrons move with the magnitude of their linear momentum having a ratio of 2: 1. If the slower electron moves with a speed of1.20 $\displaystyle \times 10^{8}\frac{m}{s}$, what is the speed of the faster moving electron?

(A) $2.67\displaystyle \times 10^{8}\frac{m}{s}$
(B) $2.40\displaystyle \times 10^{8}\frac{m}{s}$
(C) $2.24\displaystyle \times 10^{8}\frac{m}{s}$
(D) $1.97\displaystyle \times 10^{8}\frac{m}{s}$
(E) $1.56\displaystyle \times 10^{8}\frac{m}{s}$

Two clocks, A and $\mathrm{B}$, are synchronized on Earth. Clock A is placed onto a space ship that leaves Earth in a straight line with a speed of $2.40\displaystyle \times 10^{8}\frac{m}{s}$. On Earth, a scientist with clock $\mathrm{B}$ has her telescope fixed directly on clock A. If each clock started at $t=0s$, what time does the scientist observe on clock A when clock $\mathrm{B}$ reads $t=90s$? Assume the time of acceleration for the ship leaving the Earth was negligible.

(A) 24 $s$
(B) 30 $s$
(C) 50 $s$
(D) 54 $s$
(E) 72 $s$

The kinetic energy associated with an electron is twice its rest energy. At what speed is the electron traveling?

(A) $2.83\displaystyle \times 10^{8}\frac{m}{s}$
(B) $2.67\displaystyle \times 10^{8}\frac{m}{s}$
(C) $2.60\displaystyle \times 10^{8}\frac{m}{s}$
(D) $2.25\displaystyle \times 10^{8}\frac{m}{s}$
(E) $2.12\displaystyle \times 10^{8}\frac{m}{s}$

John is at rest on a platform that he measures to be 500 $m$ long on the $\mathrm{x}$-axis. He then sees a spaceship moving along the $\mathrm{x}$-axis with a speed of1.80 $\displaystyle \times 10^{8}\frac{m}{s}$ as it crosses the length of the platform. Sitting in her chair, the $\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}^{1}\mathrm{s}$ pilot measures the time it takes her to cross the platform. What type oftime interval does she measure and what value does she obtain?

Two identical particles travel to the right. The particle traveling at $2.40\displaystyle \times 10^{8}\frac{m}{s}$ collides with and sticks to the other particle traveling at $1.80\displaystyle \times 10^{8}\frac{m}{s}$. Which one of the following choices best represents the speed of the resulting object after collision?

(A) $2.20\displaystyle \times 10^{8}\frac{m}{s}$
(B) $2.14\displaystyle \times 10^{8}\frac{m}{s}$
(C) $2.10\displaystyle \times 10^{8}\frac{m}{s}$
(D) $2.06\displaystyle \times 10^{8}\frac{m}{s}$
(E) $2.00\displaystyle \times 10^{8}\frac{m}{s}$

An object of mass $m$ is initially at rest. After this object is accelerated to a speed of 2.40 $\mathrm{x}10^{8}\mathrm{m}/\mathrm{s}$, it collides with and sticks to a second object of mass $m$ that is at rest. Immediately after the collision, what is the common speed of the two masses?

(A) 2.25 $\mathrm{x}10^{8}\mathrm{m}/\mathrm{s}$
(B) 1.80 $\mathrm{x}10^{8}\mathrm{m}/\mathrm{s}$
(C) 1.66 $\mathrm{x}10^{8}\mathrm{m}/\mathrm{s}$
(D) 1.50 $\mathrm{x}10^{8}\mathrm{m}/\mathrm{s}$
(E) 1.20 $\mathrm{x}10^{8}\mathrm{m}/\mathrm{s}$

A particle has a total energy of 500 $MeV$ and a linear momentum of $300\displaystyle \frac{MeV}{c}$. What is the mass of the particle?

(A) $800\displaystyle \frac{MeV}{c^{2}}$
(B) $583\displaystyle \frac{MeV}{c^{2}}$
(C) $400\displaystyle \frac{MeV}{c^{2}}$
(D) $267\displaystyle \frac{MeV}{c^{2}}$
(E) $200\displaystyle \frac{MeV}{c^{2}}$

A lump of clay whose rest mass is 4.0 kg is traveling at three-fifths of the speed of light when it collides head-on with an identical lump going in the opposite direction at the same speed. If the two lumps stick together and no energy is radiated away, what is the mass of the composite lump?

(A) 4.0 kg
(B) 6.4 kg
(C) 8.0 kg
(D) 10.0 kg
(E) 13.3 kg

A free electron (rest mass: $m_{e}=0.5 \frac{\mathrm{MeV}}{\mathrm{c}^{2}}$ ) has a total energy of 1.5 MeV . Its momentum, $p$, in terms of $\frac{\mathrm{MeV}}{\mathrm{c}}$ is about
(A) 0.86
(B) 1.0
(C) 1.4
(D) 1.5
(E) 2.0

An electron ( $e^{-}$) is moving and has kinetic energy of 1.00 MeV . It makes a head-on collision with a positron $\left(e^{+}\right)$that is at rest. In the collision the two particles annihilate each other and are replaced by two photons ( $\gamma$ ) of equal energy, each traveling at angles $\theta$ to the electron's original direction of motion. The reaction is $\mathrm{e}^{-}+\mathrm{e}^{+} \rightarrow 2 \gamma$. Determine the angle of emission $\theta$ of each photon.
(A) $15.3^{\circ}$
(B) $25.3^{\circ}$
(C) $35.3^{\circ}$
(D) $45.3^{\circ}$
(E) $55.3^{\circ}$

The relativistic equations for time dilation, length contraction, momentum, and energy are accurate...
(A) at typical low speeds only.
(B) at speeds near the speed of light only.
(C) at all speeds up to the speed of light.
(D) at the speed of light only.
(E) at speeds above the speed of light only.

An incredibly energetic spaceship has a "proper length" of $L=200 \mathrm{~m}$, as determined by an observer on board the ship. It is traveling at a speed 0.97c relative to the Earth. Assuming the spaceship travels parallel to its length $L$, what is the length of the spaceship as measured by a physicist that observes the ship from Earth?
(A) 200 m
(B) 194 m
(C) 48.6 m
(D) 206 m
(E) 97.0 m