Question 1

A spaceship is 25 m wide and 100 m long. When traveling at 0.80 c (in a direction parallel to its length), what is its observed width?
A) 42 m
B) 25 m
C) 16 m
D) 15 m
E) 20 m

Accepted Answer

The width of the spaceship does not change because length contraction only occurs in the direction of motion. Since the spaceship is moving parallel to its length, only the length would contract, not the width.

Question 2

What error is made calculating the momentum of a particle moving at 0.80 c using classical mechanics rather than relativistic mechanics?
A) less than 1%
B) the classical value is 67% too high
C) the classical value is 67% too low
D) the classical value is 40% too high
E) the classical value is 40% too low

Accepted Answer

The answer of What error is made calculating the momentum...

Question 3

The front of a spacecraft forms a point (like the tip of a cone) with an interior opening angle of 60°. As the spacecraft increases speed relative to Earth, what happens to this angle as observed in the Earth frame?
A) It stays the same.
B) It decreases.
C) It increases.
D) It decreases only if the spacecraft is going to the left.
E) It decreases only if the spacecraft is going to the right.

Accepted Answer

The answer of The front of a spacecraft forms a...

Question 4

Two cars are traveling down the highway in the same direction. Car A is traveling at 30.0 m/s and car B is passing car A at a relative speed of 3.0 m/s. Using Galilean relativity to calculate the speed of car B relative to the highway results in an answer that is incorrect by how much? A) 3.3 × 10^-2 m/s B) 3.3 × 10^-5 m/s C) 3.3 × 10^-8 m/s D) 3.3 × 10^-11 m/s E) 3.3 × 10^-14 m/s

Accepted Answer

The answer of Two cars are traveling down the highway...

Question 5

What speed is required for a length to contract to 80% of its proper length?
A) 0.80 c
B) 0.60 c
C) 0.40 c
D) 0.20 c
E) 0.50 c

Accepted Answer

According to the Lorentz contraction formula, the length contraction (L') is given by L' = L(1-v^2/c^2)^0.5, where L is the proper length, v is the velocity of the object, and c is the speed of light. To find the velocity required for a length to contract to 80% of its proper length, we need to solve for v when L' = 0.8L. So, 0.8L = L(1-v^2/c^2)^0.5, which simplifies to (0.8)^2 = 1 - v^2/c^2. Solving for v, we get v = 0.6c. Therefore, the answer is B.

Question 6

A spaceship is observed by a stationary observer to be moving at 0.600 c. A laser beam is directed straight ahead from the front of the spaceship. How fast does the laser beam travel relative to the stationary observer?
A) 4.00 c
B) 1.60 c
C) 1.00 c
D) 0.750 c
E) 0.480 c

Accepted Answer

According to Einstein's theory of special relativity, the speed of light is constant in all reference frames. Therefore, the laser beam will travel at the speed of light relative to the stationary observer, which is 1.00 c. The spaceship's velocity does not affect the speed of the laser beam.

Question 7

What is the rest energy of a proton? (m_p = 1.67 × 10^-27 kg) A) 1.5 × 10^-10 J B) 5.0 × 10^-13 J C) 1.6 × 10^-13 J D) 1.6 × 10^-19 J E) 0.0

Accepted Answer

The answer of What is the rest energy of a...

Question 8

A spear is thrown at a very high speed. As it passes, you measure its length to be half its normal length. From this you conclude that the spear must be moving at
A) 1.00 c
B) 0.866 c
C) 0.707 c
D) 0.500 c
E) Not enough information

Accepted Answer

According to the theory of relativity, the length contraction formula is $L = L_0\sqrt{1 - \frac{v^2}{c^2}}$, where $L$ is the observed length, $L_0$ is the rest length, $v$ is the velocity of the object, and $c$ is the speed of light. If the observed length $L$ is half of the rest length $L_0$, then we have $0.5 = \sqrt{1 - \frac{v^2}{c^2}}$. Squaring both sides gives $0.25 = 1 - \frac{v^2}{c^2}$, which simplifies to $\frac{v^2}{c^2} = 0.75$. Taking the square root of both sides gives $\frac{v}{c} = \sqrt{0.75} = 0.866$, meaning the spear must be moving at 0.866 times the speed of light.

Question 9

In a colliding-beam experiment, protons moving at 0.800 c in opposite directions strike one another. What is the speed of the incoming proton as viewed by the proton about to be hit?
A) 1.60 c
B) 0.00
C) 0.640 c
D) 0.976 c
E) 0.960 c

Accepted Answer

When two objects move towards each other, their relative speed is the sum of their individual speeds. In this case, the speed of each proton is 0.800 c. Therefore, the speed of the incoming proton as viewed by the proton about to be hit is:

v = 0.800 c + 0.800 c
v = 1.600 c

However, according to the principle of relativity, the speed of an object cannot be greater than the speed of light (c). Therefore, we need to use the relativistic velocity addition formula:

v = (v1 + v2) / (1 + v1v2/c^2)

where v1 = v2 = 0.8c (since the protons have the same speed), and c is the speed of light.

v = (0.8c + 0.8c) / (1 + 0.8c x 0.8c/c^2)
v = 1.6c / (1 + 0.64)
v = 1.6c / 1.64
v = 0.976c

Therefore, the speed of the incoming proton as viewed by the proton about to be hit is 0.976 c, which is closest to answer choice D.

Question 10

Two cars are traveling down the highway in the same direction. Car A is traveling at 30.0 m/s and car B is passing car A at a relative speed of 3.0 m/s. What is the speed of car B with respect to the highway?
A) 27.0 m/s
B) 30.0 m/s
C) 33.0 m/s
D) 32.9 m/s
E) 33.1 m/s

Accepted Answer

The speed of car B with respect to the highway is equal to its speed relative to car A plus the speed of car A with respect to the highway. Therefore, the speed of car B with respect to the highway is 30.0 m/s (the speed of car A with respect to the highway) + 3.0 m/s (the speed of car B relative to car A) = 33.0 m/s.

Question 11

If two protons are heading toward one another at 0.800 c in the laboratory frame, what is the kinetic energy of one proton in the frame of reference of the other proton? The mass of a proton is 938 MeV/c². A) 3.13 GeV B) 5.56 GeV C) 1.56 GeV D) 2.50 GeV E) 3.34 GeV

Accepted Answer

The answer of If two protons are heading toward one...

Question 12

If an object has its speed doubled, what happens to its relativistic momentum?
A) Nothing happens.
B) The momentum doubles.
C) The momentum halves.
D) The momentum more than doubles.
E) The momentum increases a little less than double.

Accepted Answer

When an object's speed is doubled, its relativistic momentum increases more than double according to the formula p=mv/√(1-v^2/c^2). As the speed increases, the denominator in the formula decreases, leading to a greater value of p. Therefore, the momentum more than doubles.

Question 13

Spaceships A and B are moving in the same direction. Spaceship A is moving at 0.80 c relative to a stationary observer, and spaceship B is moving at 0.80 c relative to spaceship A. What does the stationary observer find for the speed of spaceship B?
A) 1.6 c
B) 1.3 c
C) 1.0 c
D) 0.78 c
E) 0.98 c

Accepted Answer

To find the relative speed of spaceship B as observed by the stationary observer, we use the formula for relativistic addition of velocities: $v = \frac{v_1 + v_2}{1 + \frac{v_1 \cdot v_2}{c^2}}$, where $v_1 = 0.80c$ (speed of A relative to the observer), $v_2 = 0.80c$ (speed of B relative to A), and $c$ is the speed of light. Plugging in the values, we get $v = \frac{0.80c + 0.80c}{1 + \frac{0.80c \cdot 0.80c}{c^2}} = \frac{1.6c}{1 + 0.64} = \frac{1.6c}{1.64} \approx 0.98c$. This shows that due to the effects of special relativity, speeds don't simply add up linearly when approaching the speed of light.

Question 14

A spaceship is observed from the Earth to be moving toward a star at a speed of 0.80 c. If the distance from the spaceship to the star is 1.6 ly as measured from Earth, how far is the star as measured from the moving spaceship?
A) 2.7 ly
B) 1.9 ly
C) 1.6 ly
D) 1.2 ly
E) 0.96 ly

Accepted Answer

The correct answer is E.The distance from the spaceship to the star is 1.6 ly as measured from Earth. The spaceship is moving toward the star at a speed of 0.80 c. The distance from the spaceship to the star as measured from the moving spaceship is given by the following equation:$$d' = d/\gamma$$where:* d' is the distance from the spaceship to the star as measured from the moving spaceship* d is the distance from the spaceship to the star as measured from Earth* γ is the Lorentz factorThe Lorentz factor is given by the following equation:$$\gamma = 1/\sqrt{1 - v^2/c^2}$$where:* v is the speed of the spaceship* c is the speed of lightSubstituting the given values into the equations, we get:$$d' = 1.6 ly / 1.667 = 0.96 ly$$

Question 15

In the lab, charged particles are directed toward one another, each at speed 0.500 c. What is the speed of the particles moving one way with respect to the particles moving the other way?
A) 1.25 c
B) 1.00 c
C) 0.800 c
D) 0.500 c
E) 0

Accepted Answer

The answer of In the lab, charged particles are directed...

Question 16

If a proton has a kinetic energy of 1.000 GeV, what is its momentum? (m_p = 938.3 MeV/c²) A) 1.938 GeV/c B) 1.000 GeV/c C) 938.0 GeV/c D) 2.872 GeV/c E) 1.696 GeV/c

Accepted Answer

The momentum of a particle can be calculated using the formula p = sqrt(E^2 - m^2c^4)/c, where E is the kinetic energy, m is the mass of the particle, and c is the speed of light.

Substituting the values given in the question, we get:

p = sqrt((1.000 GeV)^2 - (938.3 MeV/c^2)^2)/c

= sqrt(1.000^2 - 0.880^2) * (10^9 eV / 3 * 10^8 m/s)

= 1.696 GeV/c

Therefore, the correct answer is E.

Question 17

What is the magnitude of the momentum of a proton moving at 0.95 c? (m_p = 1.67 × 10^-27 kg) A) 5.0 × 10^-18 kg·m/s B) 4.8 × 10^-18 kg·m/s C) 1.5 × 10^-18 kg·m/s D) 9.5 × 10^-18 kg·m/s E) 4.4 × 10^-18 kg·m/s

Accepted Answer

The answer of What is the magnitude of the momentum...

Question 18

Event 2 occurs a time Δt after event 1. If event 2 is caused by event 1, which of the following is true?
A) The distance between event 1 and event 2 can be anything.
B) The distance between event 1 and event 2 is not more than c Δt.
C) There are frames of reference in which event 2 occurs before event 1.
D) There are frames of reference in which event 1 and event 2 are simultaneous.
E) None of these choices are correct.

Accepted Answer

According to special relativity, the maximum speed at which information can be transmitted is the speed of light, c. Therefore, if event 2 is causally linked to event 1, it cannot occur more than c Δt away from event 1 in any frame of reference. Choices A, C, D, and E are incorrect because they either do not follow from the principles of special relativity or contradict them.

Question 19

If an object increases its speed such that its total energy is doubled, what happens to its relativistic momentum?
A) The momentum is doubled.
B) The momentum is more than doubled.
C) The momentum is less than doubled.
D) The momentum is quadrupled.
E) Nothing happens to the momentum.

Accepted Answer

According to the equation for total relativistic energy (E=γmc^2), if the energy is doubled, then the factor γ (Lorentz factor) must also double. This means that the velocity of the object is increased, and therefore its relativistic momentum (p=γmv) is also increased. Since the momentum is proportional to the velocity, it will be more than doubled.

Question 20

If a spaceship of proper length 40 m is measured to have a length 30 m, how fast is it moving?
A) 0.50 c
B) 0.81 c
C) 0.75 c
D) 0.66 c
E) 0.43 c

Accepted Answer

The answer of If a spaceship of proper length 40...

Quiz 26: Relativity

A spaceship is 25 m wide and 100 m long. When traveling at 0.80 c (in a direction parallel to its length) , what is its observed width?

What error is made calculating the momentum of a particle moving at 0.80 c using classical mechanics rather than relativistic mechanics?

The front of a spacecraft forms a point (like the tip of a cone) with an interior opening angle of 60°. As the spacecraft increases speed relative to Earth, what happens to this angle as observed in the Earth frame?

Two cars are traveling down the highway in the same direction. Car A is traveling at 30.0 m/s and car B is passing car A at a relative speed of 3.0 m/s. Using Galilean relativity to calculate the speed of car B relative to the highway results in an answer that is incorrect by how much?

What speed is required for a length to contract to 80% of its proper length?

A spaceship is observed by a stationary observer to be moving at 0.600 c. A laser beam is directed straight ahead from the front of the spaceship. How fast does the laser beam travel relative to the stationary observer?

What is the rest energy of a proton? (m_p = 1.67 × 10^-27 kg)

A spear is thrown at a very high speed. As it passes, you measure its length to be half its normal length. From this you conclude that the spear must be moving at

In a colliding-beam experiment, protons moving at 0.800 c in opposite directions strike one another. What is the speed of the incoming proton as viewed by the proton about to be hit?

Two cars are traveling down the highway in the same direction. Car A is traveling at 30.0 m/s and car B is passing car A at a relative speed of 3.0 m/s. What is the speed of car B with respect to the highway?

If two protons are heading toward one another at 0.800 c in the laboratory frame, what is the kinetic energy of one proton in the frame of reference of the other proton? The mass of a proton is 938 MeV/c².

If an object has its speed doubled, what happens to its relativistic momentum?

Spaceships A and B are moving in the same direction. Spaceship A is moving at 0.80 c relative to a stationary observer, and spaceship B is moving at 0.80 c relative to spaceship A. What does the stationary observer find for the speed of spaceship B?

A spaceship is observed from the Earth to be moving toward a star at a speed of 0.80 c. If the distance from the spaceship to the star is 1.6 ly as measured from Earth, how far is the star as measured from the moving spaceship?

In the lab, charged particles are directed toward one another, each at speed 0.500 c. What is the speed of the particles moving one way with respect to the particles moving the other way?

If a proton has a kinetic energy of 1.000 GeV, what is its momentum? (m_p = 938.3 MeV/c²)

What is the magnitude of the momentum of a proton moving at 0.95 c? (m_p = 1.67 × 10^-27 kg)

Event 2 occurs a time Δt after event 1. If event 2 is caused by event 1, which of the following is true?

If an object increases its speed such that its total energy is doubled, what happens to its relativistic momentum?

If a spaceship of proper length 40 m is measured to have a length 30 m, how fast is it moving?