Question 1

Lorentz transformation: System S' has a velocity u = +0.56c relative to system S, as shown in the figure. The clocks of S and S' are synchronized at t = t' = 0 when the origins O and O' coincide. An event is observed in both systems. The event takes place at x = 800 m and at time t = 3.0 μs as measured by an observer in S. What is the time of the event, measured by an observer in S'?

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Question 2

Kinetic energy: In a certain particle accelerator, a proton has a kinetic energy that is equal to its rest energy. What is the speed of the proton relative to the accelerator?

Accepted Answer

When the kinetic energy (K) of a particle equals its rest energy (E₀), the total energy (E) is given by E = E₀ + K = 2E₀. According to the theory of relativity, the total energy E is related to the rest mass m₀ and the speed v by the equation E = γm₀c², where γ = 1/√(1 - v²/c²) and c is the speed of light. Setting E = 2E₀ implies γ = 2, which leads to 1/√(1 - v²/c²) = 2. Solving for v/c gives v/c = √3/2, which is approximately 0.87. Therefore, the speed of the proton relative to the accelerator is approximately 0.87c.

Question 3

Velocity addition: Consider three galaxies, Alpha, Beta and Gamma. An observer in Beta sees the other two galaxies each moving away from him in opposite directions at speed 0.70c. At what speed would an observer in Alpha see the galaxy Beta moving?

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Question 4

Velocity addition: A spaceship is moving away from the earth with a constant speed of 0.80c. The spaceship fires a 28-kg missile with a speed of 0.50c relative to the spaceship. What is the speed of the missile measured by observers on the earth if the missile is fired
(a) away from the earth?
(b) toward the earth?

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Question 5

Momentum: A particle is moving at 0.75c relative to a lab on Earth. By what percentage is the Newtonian expression for its momentum in error? (The percentage error is the difference between the erroneous and correct values, divided by the correct one).

Accepted Answer

The Newtonian expression for momentum is p = mv, but at relativistic speeds, the correct expression is p = γmv, where γ is the Lorentz factor. At 0.75c, γ ≈ 1.511. The percentage error is (γ - 1)/γ = (1.511 - 1)/1.511 ≈ 0.338, or 33.8%, which rounds to 34%.

Question 6

Velocity addition: The captain of spaceship A observes enemy spaceship E escaping with a relative velocity of 0.48c, as shown in the figure. A missile M is fired from ship A, with a velocity of 0.72c relative to ship A. What is the relative velocity of approach of missile M, observed by the crew on ship E?

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Question 7

Doppler effect: A navigational beacon in deep space broadcasts at a radio frequency of 50 MHz. A spaceship approaches the beacon with a relative velocity of 0.40c. The spaceship passes the beacon and departs from it with a DIFFERENT relative velocity. The beacon signal is now detected at a frequency of 40 MHz. What is the new velocity of the spaceship relative to the beacon?

Accepted Answer

Using the formula for Doppler effect:

$f_o = f_s \frac{c + v}{c - v}$

where:
$f_o$ = observed frequency
$f_s$ = source frequency
$c$ = speed of light
$v$ = relative velocity between source and observer

When the spaceship approaches the beacon:

$f_o = 40	ext{ MHz}$
$f_s = 50	ext{ MHz}$
$c = 3	imes10^8 	ext{ m/s}$
$v = 0.4c$

Solving for $v$:

$40	ext{ MHz} = 50	ext{ MHz} \frac{3	imes10^8 	ext{ m/s} + v}{3	imes10^8 	ext{ m/s} - v}$
$0.8 = \frac{1.4c}{1.6c - v}$
$1.12c - 0.8v = 1.4c$
$0.6v = 0.28c$
$v = 0.4667c$

When the spaceship departs the beacon, the observed frequency is still 40 MHz. Solving for the new velocity:

$f_o = 40	ext{ MHz}$
$f_s = 50	ext{ MHz}$
$c = 3	imes10^8 	ext{ m/s}$
$v = ?$

$40	ext{ MHz} = 50	ext{ MHz} \frac{3	imes10^8 	ext{ m/s} + v}{3	imes10^8 	ext{ m/s} - v}$
$0.8 = \frac{1.4c}{1.6c + v}$
$1.12c + 0.8v = 1.4c$
$0.8v = 0.28c$
$v = 0.35c$

Therefore, the new velocity of the spaceship relative to the beacon is:

$v_{new} = 0.35c$

The best choice is B because it matches the calculated value.

Question 8

Momentum: In the lab, a relativistic proton has a momentum of 1.00 × 10^-19 kg ∙ m/s and a rest energy of 0.150 nJ. What is the speed of the proton in the lab? (c = 3.00 × 10⁸ m/s, m_proton = 1.67 × 10^-27 kg)

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Question 9

Kinetic energy: An electron is accelerated from rest through a potential difference of 50.0 kV. What is the speed of the electron? (c = 3.00 × 10⁸ m/s, e = 1.60 × 10^-19 C, m_el = 9.11 × 10^-31 kg)

Accepted Answer

The kinetic energy gained by the electron is equal to the work done on it by the electric field, which is eV. Using the equation for kinetic energy (1/2)mv² = eV, we can solve for v: v = sqrt(2eV/m). Substituting the given values, v = sqrt(2 * 1.60 × 10^-19 C * 50.0 × 10^3 V / 9.11 × 10^-31 kg) ≈ 1.24 × 10^8 m/s.

Question 10

Velocity addition: In an "atom smasher," two particles collide head on at relativistic speeds. If the velocity of the first particle is 0.741c to the left, and the velocity of the second particle is  to the right (both of these speeds are measured in Earth's rest frame), how fast are the particles moving with respect to each other?

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Question 11

Velocity addition: A spaceship approaching an asteroid at a speed of 0.60c launches a rocket forward with a speed of 0.40c relative to the spaceship. At what speed is the rocket approaching the asteroid as measured by an astronaut on the asteroid?

Accepted Answer

The speed of the rocket relative to the asteroid can be calculated using the formula for relativistic velocity addition: $v = \frac{v_1 + v_2}{1 + \frac{v_1v_2}{c^2}}$, where $v_1 = 0.60c$ and $v_2 = 0.40c$. Plugging these values in, we get $v = \frac{0.60c + 0.40c}{1 + \frac{(0.60c)(0.40c)}{c^2}} = \frac{1.0c}{1 + 0.24} = \frac{1.0c}{1.24} = 0.81c$.

Question 12

Length contraction: Two space stations are at rest relative to each other and are 6.0 × 10⁷ m apart, as measured by observers on the stations. A spaceship traveling from one station to the other at 0.90c relative to the stations passes both of them, one after the other. As measured by an observer in the spaceship, how long does it take to travel from one station to the other? (c = 3.00 × 10⁸ m/s)

Accepted Answer

Explanation:The length contraction formula is:$$L = L_0/\gamma$$where:L is the length of the object as measured by an observer in the moving frameL_0 is the length of the object as measured by an observer in the rest frameγ is the Lorentz factor, which is given by:$$\gamma = 1/\sqrt{1 - v^2/c^2}$$In this problem, the length of the object is 6.0 × 10^7 m, and the speed of the spaceship is 0.90c. Therefore, the Lorentz factor is:$$\gamma = 1/\sqrt{1 - 0.90^2} = 2.29$$The length of the object as measured by an observer in the spaceship is:$$L = 6.0 × 10^7 m/2.29 = 2.62 × 10^7 m$$The time it takes for the spaceship to travel from one station to the other is:$$t = L/v = (2.62 × 10^7 m)/(0.90 × 3.00 × 10^8 m/s) = 97 ms$$Therefore, the correct answer is A.

Question 13

Velocity addition: Two spaceships are approaching one another, each at a speed of 0.28c relative to a stationary observer on Earth. What speed does an observer on one spaceship record for the other approaching spaceship?

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Question 14

Momentum: At what speed relative to the lab will a 0.272-kg object have the same momentum as a 1.30-kg object that is moving at 0.515c relative to the lab?

Accepted Answer

The momentum of an object is given by $p = mv$, where $m$ is the mass and $v$ is the velocity. For the two objects to have the same momentum, their $mv$ products must be equal. Setting the momentum of the two objects equal to each other gives $0.272 \times v = 1.30 \times 0.515c$. Solving for $v$ gives $v = \frac{1.30 \times 0.515c}{0.272} = 0.944c$.

Question 15

Lorentz transformation: System S' has a velocity u = +0.45c relative to system S, as shown in the figure. The clocks of S and S' are synchronized at t = t' = 0 when the origins O and O' coincide. An event is observed in both systems. The event takes place at x = 600 m and at time t = 1.9 μs, as measured by an observer in S. What is the x'-coordinate of the event, measured by an observer in S'?

Accepted Answer

Using the Lorentz transformation formula for the x coordinate:

$x'=\gamma(x-ut)$

where $\gamma=\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}$ is the Lorentz factor.

Plugging in the given values, we get:

$x'=\gamma(x-ut)=\frac{1}{\sqrt{1-\frac{(0.45c)^2}{c^2}}}(600	ext{ m}-0.45c	imes1.9	ext{ μs})\approx 380	ext{ m}$

Therefore, the x'-coordinate of the event, measured by an observer in S', is 380 m.

Question 16

Length contraction: A spacecraft is measured by an observer on the ground to have a length of 53 m as it flies toward the earth with a speed 1.7 × 10⁸ m/s. The spacecraft then lands and its length is again measured by the observer on the ground, this time while the spacecraft is at rest on the launchpad. What result does he now get for the length? (c = 3.00 × 10⁸ m/s)

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Question 17

Velocity addition: Three spaceships A, B, and C are in motion, as shown in the figure. The commander on ship B observes ship C approaching with a relative velocity of 0.78c. The commander also observes ship A, advancing in the rear, with a relative velocity of 0.31c. What is the velocity of ship C, relative to an observer on ship A?

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Question 18

Kinetic energy: An electron has a speed of 0.643c. Through what potential difference would the electron need to be accelerated (starting from rest) in order to reach this speed? (c = 3.00 × 10⁸ m/s, e = 1.60 × 10^-19 C, m_el = 9.11 × 10^-31 kg)

Accepted Answer

The kinetic energy of the electron can be calculated using the equation:

KE = (1/2)mv²

where KE is the kinetic energy, m is the mass of the electron, and v is the speed of the electron.

Substituting the given values, we get:

KE = (1/2)(9.11 × 10^-31 kg)(0.643c)^2

KE = 3.02 × 10^-14 J

The potential difference can be calculated using the equation:

PE = qΔV

where PE is the potential energy gained by the electron, q is the charge of the electron, and ΔV is the potential difference.

Equating the kinetic energy and potential energy gained by the electron, we get:

(1/2)mv² = qΔV

Rearranging the equation, we get:

ΔV = (1/2)(mv²/q)

Substituting the given values, we get:

ΔV = (1/2)(9.11 × 10^-31 kg)(0.643c)^2/(1.60 × 10^-19 C)

ΔV = 160 kV

Therefore, the potential difference needed to accelerate the electron to a speed of 0.643c is 160 kV.

Question 19

Doppler effect: A navigational beacon in deep space broadcasts at a radio frequency of 50 MHz. A spaceship approaches the beacon with a relative velocity of 0.40c. What is the frequency of the beacon radio signal that is detected on the ship?

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Question 20

Velocity addition: Three spaceships A, B, and C are in motion as shown in the figure. The commander on ship B observes ship C approaching with a relative velocity of 0.83c. The commander also observes ship A, advancing in the rear, with a relative velocity of 0.48c. As measured by commander on ship B, at what speed is ship A approaching ship C?

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Quiz 29: Electromagnetic Induction

Kinetic energy: In a certain particle accelerator, a proton has a kinetic energy that is equal to its rest energy. What is the speed of the proton relative to the accelerator?

Velocity addition: Consider three galaxies, Alpha, Beta and Gamma. An observer in Beta sees the other two galaxies each moving away from him in opposite directions at speed 0.70c. At what speed would an observer in Alpha see the galaxy Beta moving?

Momentum: A particle is moving at 0.75c relative to a lab on Earth. By what percentage is the Newtonian expression for its momentum in error? (The percentage error is the difference between the erroneous and correct values, divided by the correct one) .

Momentum: In the lab, a relativistic proton has a momentum of 1.00 × 10^-19 kg ∙ m/s and a rest energy of 0.150 nJ. What is the speed of the proton in the lab? (c = 3.00 × 10⁸ m/s, m_proton = 1.67 × 10^-27 kg)

Kinetic energy: An electron is accelerated from rest through a potential difference of 50.0 kV. What is the speed of the electron? (c = 3.00 × 10⁸ m/s, e = 1.60 × 10^-19 C, m_el = 9.11 × 10^-31 kg)

Velocity addition: A spaceship approaching an asteroid at a speed of 0.60c launches a rocket forward with a speed of 0.40c relative to the spaceship. At what speed is the rocket approaching the asteroid as measured by an astronaut on the asteroid?

Velocity addition: Two spaceships are approaching one another, each at a speed of 0.28c relative to a stationary observer on Earth. What speed does an observer on one spaceship record for the other approaching spaceship?

Momentum: At what speed relative to the lab will a 0.272-kg object have the same momentum as a 1.30-kg object that is moving at 0.515c relative to the lab?

Kinetic energy: An electron has a speed of 0.643c. Through what potential difference would the electron need to be accelerated (starting from rest) in order to reach this speed? (c = 3.00 × 10⁸ m/s, e = 1.60 × 10^-19 C, m_el = 9.11 × 10^-31 kg)

Doppler effect: A navigational beacon in deep space broadcasts at a radio frequency of 50 MHz. A spaceship approaches the beacon with a relative velocity of 0.40c. What is the frequency of the beacon radio signal that is detected on the ship?

Quiz 29: Electromagnetic Induction

Kinetic energy: In a certain particle accelerator, a proton has a kinetic energy that is equal to its rest energy. What is the speed of the proton relative to the accelerator?

Velocity addition: Consider three galaxies, Alpha, Beta and Gamma. An observer in Beta sees the other two galaxies each moving away from him in opposite directions at speed 0.70c. At what speed would an observer in Alpha see the galaxy Beta moving?

Momentum: A particle is moving at 0.75c relative to a lab on Earth. By what percentage is the Newtonian expression for its momentum in error? (The percentage error is the difference between the erroneous and correct values, divided by the correct one) .

Momentum: In the lab, a relativistic proton has a momentum of 1.00 × 10-19 kg ∙ m/s and a rest energy of 0.150 nJ. What is the speed of the proton in the lab? (c = 3.00 × 108 m/s, mproton = 1.67 × 10-27 kg)

Kinetic energy: An electron is accelerated from rest through a potential difference of 50.0 kV. What is the speed of the electron? (c = 3.00 × 108 m/s, e = 1.60 × 10-19 C, mel = 9.11 × 10-31 kg)

Velocity addition: A spaceship approaching an asteroid at a speed of 0.60c launches a rocket forward with a speed of 0.40c relative to the spaceship. At what speed is the rocket approaching the asteroid as measured by an astronaut on the asteroid?

Velocity addition: Two spaceships are approaching one another, each at a speed of 0.28c relative to a stationary observer on Earth. What speed does an observer on one spaceship record for the other approaching spaceship?

Momentum: At what speed relative to the lab will a 0.272-kg object have the same momentum as a 1.30-kg object that is moving at 0.515c relative to the lab?

Kinetic energy: An electron has a speed of 0.643c. Through what potential difference would the electron need to be accelerated (starting from rest) in order to reach this speed? (c = 3.00 × 108 m/s, e = 1.60 × 10-19 C, mel = 9.11 × 10-31 kg)

Doppler effect: A navigational beacon in deep space broadcasts at a radio frequency of 50 MHz. A spaceship approaches the beacon with a relative velocity of 0.40c. What is the frequency of the beacon radio signal that is detected on the ship?

Momentum: In the lab, a relativistic proton has a momentum of 1.00 × 10^-19 kg ∙ m/s and a rest energy of 0.150 nJ. What is the speed of the proton in the lab? (c = 3.00 × 10⁸ m/s, m_proton = 1.67 × 10^-27 kg)

Kinetic energy: An electron is accelerated from rest through a potential difference of 50.0 kV. What is the speed of the electron? (c = 3.00 × 10⁸ m/s, e = 1.60 × 10^-19 C, m_el = 9.11 × 10^-31 kg)

Kinetic energy: An electron has a speed of 0.643c. Through what potential difference would the electron need to be accelerated (starting from rest) in order to reach this speed? (c = 3.00 × 10⁸ m/s, e = 1.60 × 10^-19 C, m_el = 9.11 × 10^-31 kg)