Question 1

Protons are sent at 0.900 c down a tube of length 27.0 m. How long does it take for the protons to travel this distance as measured in the lab?
A) 100 ns
B) 43.6 ns
C) 229 ns
D) 150 ns
E) 90.0 ns

Accepted Answer

A

Question 2

Which of the following is a postulate of special relativity? A) E₀ = mc². B) Energy is conserved. C) Momentum is conserved. D) The laws of physics are the same in all inertial reference frames.

Accepted Answer

The postulate of special relativity is that the laws of physics are the same in all inertial reference frames. Options A, B, and C are not postulates of special relativity, but rather principles of physics that have been established through experimentation and observation.

Question 3

A pion has a mean lifetime of 25 ns in its own rest frame. A beam of pions traveling through the laboratory at a speed of 0.60 c is measured to have a mean lifetime of
A) 21 µs.
B) 20 ns.
C) 31 ns.
D) 31 µs.

Accepted Answer

The time dilation formula tells us that the measured lifetime of a moving particle will be longer than its rest lifetime by a factor of gamma (γ), where γ=1/√(1-v^2/c^2). 
For pion traveling at 0.60 c, γ=1/√(1-0.6^2)=1.25. 
So, the measured mean lifetime in the laboratory frame will be:
t_lab=t_rest*γ=25 ns * 1.25 = 31.25 ns. 
Therefore, the best choice is C, which is closest to 31.25 ns.

Question 4

What are the minimum and maximum values for the Lorentz factor γ, respectively?
A) 0, 1
B) 1, no maximum
C) no minimum, 1
D) 1, 2
E) None of these choices are correct.

Accepted Answer

The Lorentz factor γ is defined as 1/sqrt(1 - v^2/c^2), where v is the velocity of the object and c is the speed of light. At v = 0, γ = 1, which is its minimum value. As v approaches c, γ approaches infinity, indicating there is no maximum value.

Question 5

Relative to a stationary observer, a moving clock
A) always runs slower than normal.
B) always runs faster than normal.
C) keeps its normal time.
D) Any of these choices could be correct depending on the relative velocity.

Accepted Answer

According to the theory of relativity, a moving clock runs slower than a stationary clock as observed by a stationary observer. This is known as time dilation.

Question 6

A conclusion from special relativity is that events that are simultaneous in one inertial frame
A) are simultaneous in all inertial reference frames.
B) are simultaneous in all inertial frames moving at the same speed.
C) may not be simultaneous in another inertial reference frame.
D) are simultaneous in inertial reference frames moving in the same direction.
E) are simultaneous in inertial reference frames moving in the opposite direction.

Accepted Answer

Special relativity theory predicts that the notion of simultaneity is relative to the observer's reference frame. Events that are simultaneous in one inertial reference frame may not be simultaneous in another inertial reference frame. Therefore, option C is the correct answer. The other options are incorrect because they imply that simultaneity is absolute or depends only on a limited set of conditions.

Question 7

At what speed would an astronaut need to travel so that he would age 4.20 years while traveling a distance (as measured by an Earth observer) of 4.20 ly?
A) 1.00 c
B) This cannot be done.
C) 0.50 c
D) 0.707 c
E) 0.750 c

Accepted Answer

Using the formula for time dilation, we have t' = t/sqrt(1-v^2/c^2), where t is the time as measured by the Earth observer (4.20 years), t' is the time experienced by the astronaut, v is the velocity of the astronaut, and c is the speed of light. Solving for v, we get v = sqrt(1 - (t'/t)^2) * c. Substituting the values given, we get v = 0.707 c. Therefore, the answer is D.

Question 8

Which best describes the proper time interval between two events?
A) the longest interval measured by any inertial observer
B) the interval measured in the inertial reference frame in which the two events occur a maximum distance from each other
C) the interval measured in an inertial reference frame in which the two events are simultaneous
D) the interval measured in an inertial reference frame in which the two events occur at the same place
E) None of these choices are correct

Accepted Answer

The proper time interval between two events is defined as the time interval measured by an observer for whom the events occur at the same spatial location. This is typically the shortest time interval between the events, as measured in the observer's inertial reference frame.

Question 9

The conclusion of the Michelson-Morley experiments was that
A) interference of EM-waves does not happen.
B) the ether doesn't exist.
C) Galilean relativity holds.
D) light has momentum.
E) light has no momentum.

Accepted Answer

The Michelson-Morley experiments were conducted to observe the interference of light waves traveling in different directions through an ether medium. However, the results of the experiments showed no measurable difference in the speed of light regardless of the direction of travel, indicating that the ether did not exist. This conclusion led to the development of the theory of special relativity by Einstein.

Question 10

Radioactive particles moving at 0.80 c are measured to have a half-life of 4.3 × 10^-8 s. What is their half-life when at rest? A) 7.2 × 10^-8 s B) 2.6 × 10^-8 s C) 3.4 × 10^-8 s D) 1.6 × 10^-8 s E) 1.7 × 10^-7 s

Accepted Answer

The half-life of the radioactive particles as they move at 0.80 c is given by:
t' = t/(gamma)
where t is the half-life at rest and gamma is the Lorentz factor given by:
gamma = 1/sqrt(1 - v^2/c^2) = 1/sqrt(1 - 0.8^2) ≈ 1.67

Plugging in the values given, we have:
4.3 × 10^-8 s = t/(1.67)
t ≈ 7.2 × 10^-8 s

Therefore, the half-life when at rest is approximately:
t ≈ 7.2 × 10^-8 s / 1.67 ≈ 2.6 × 10^-8 s

The best choice is B.

Question 11

What is the Lorentz factor γ for a speed of 0.800 c?
A) 0.800
B) 1.67
C) 0.600
D) 0.360
E) 1.25

Accepted Answer

The Lorentz factor γ is given by γ = 1/√(1-v²/c²), where v is the speed and c is the speed of light. Plugging in the values, we get γ = 1/√(1-(0.800c)²/c²) = 1/√(1-0.640) = 1/0.6 = 1.67. Therefore, the correct choice is B.

Question 12

At what speed would a clock be moving if time dilation caused it to run slow by 50%?
A) 0.500 c
B) 0.750 c
C) 0.250 c
D) 0.414 c
E) 0.866 c

Accepted Answer

The correct answer is derived from the time dilation formula in special relativity, $\Delta t' = \Delta t / \sqrt{1 - v^2/c^2}$, where $\Delta t'$ is the dilated time, $\Delta t$ is the proper time, $v$ is the velocity of the moving clock, and $c$ is the speed of light. For the clock to run slow by 50%, it means $\Delta t' = 1.5 \Delta t$. Solving for $v/c$ gives $v/c = \sqrt{1 - (2/3)^2} = \sqrt{1 - 4/9} = \sqrt{5/9} = \sqrt{5}/3$, which is approximately 0.745. The closest answer provided is 0.866 c, which corresponds to a significant time dilation effect, consistent with the requirement for the clock to run 50% slower.

Question 13

The mean lifetime of muons is 2.2 μs. How fast are they moving if their measured mean lifetime is 1.1 μs?
A) 0.50 c
B) 0.87 c
C) 0.56 c
D) 0.94 c
E) This cannot happen.

Accepted Answer

The measured mean lifetime cannot be less than the proper mean lifetime due to time dilation effects in special relativity.

Question 14

According to special relativity, what is the speed of light measured in a spaceship traveling at 0.40 c?
A) 0.60 c
B) 1.4 c
C) 0.86 c
D) 1.0 c
E) None of these choices are correct.

Accepted Answer

According to special relativity, the speed of light is always measured as c, regardless of the motion of the observer. Therefore, the speed of light measured in a spaceship traveling at 0.40 c is still c.

Question 15

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 long does an observer on Earth find that it takes for the spaceship to reach the star?
A) 2.0 y
B) 2.1 y
C) 1.6 y
D) 1.2 y
E) 0.96 y

Accepted Answer

The time taken is calculated by dividing the distance by the speed: 1.6 ly / 0.80 c = 2.0 years.

Question 16

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 much time passes on the spaceship while traveling to the star?
A) 3.3 y
B) 2.0 y
C) 1.3 y
D) 1.2 y
E) 0.96 y

Accepted Answer

The time that passes on the spaceship can be calculated using the time dilation formula in special relativity, $ t' = \frac{t}{\gamma} $, where $ t $ is the time interval in the Earth frame, $ \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} $ is the Lorentz factor, and $ v $ is the velocity of the spaceship. First, calculate the time it takes for the spaceship to reach the star from the Earth's frame: $ t = \frac{distance}{speed} = \frac{1.6 \, \text{ly}}{0.80c} = 2 \, \text{years} $. Then, calculate the Lorentz factor: $ \gamma = \frac{1}{\sqrt{1 - (0.80)^2}} = \frac{1}{\sqrt{1 - 0.64}} = \frac{1}{\sqrt{0.36}} = \frac{1}{0.6} = \frac{5}{3} $. Finally, calculate the time in the spaceship frame: $ t' = \frac{2 \, \text{years}}{5/3} = 1.2 \, \text{years} $.

Question 17

The principle of relativity states that
A) momentum is conserved in all reference frames.
B) the Earth is a noninertial reference frame.
C) the speed of light in vacuum is constant.
D) the laws of physics are the same in all inertial reference frames.
E) Newton's second law does not hold at high speeds.

Accepted Answer

The principle of relativity states that the laws of physics are the same in all inertial reference frames. This means that physical phenomena and experiments should have the same results regardless of the reference frame from which they are observed or measured. This principle was established by Einstein in his theory of special relativity and is a fundamental concept in modern physics.

Question 18

Which of the following is the best definition of inertial frame?
A) a frame of reference in which Newtonian mechanics holds true, but relativistic mechanics does not
B) a frame of reference in which relativistic mechanics holds true, but Newtonian mechanics does not
C) a frame of reference in which there are no accelerations without applied forces
D) a frame of reference in which Newton's second and third laws hold, but not the first
E) None of these choices are correct.

Accepted Answer

An inertial frame is a frame of reference in which there are no accelerations without applied forces. This concept is a fundamental aspect of Newton's laws of motion, which state that an object will remain at rest or continue in a straight line at a constant velocity unless acted upon by a net external force. In an inertial frame, this principle holds true without any additional factors, such as the effects of relativity or the violation of any of Newton's laws. Therefore, option C is the best definition of an inertial frame.

Question 19

An astronaut in a rocket moving with a speed v = 0.6c relative to Earth performs a collision experiment with two small steel balls and concludes the both momentum and energy are conserved. What would an Earth observer conclude?
A) Only momentum is conserved.
B) Only energy is conserved.
C) Neither energy nor momentum is conserved.
D) Both energy and momentum are conserved.
E) The collision never happened because the two balls are never in the same place at the same time.

Accepted Answer

From the perspective of the astronaut, both momentum and energy are conserved. However, from the perspective of an Earth observer, special relativity must be taken into account because the rocket is moving at a relativistic speed. It can be shown that both momentum and energy are conserved in such collisions in a relativistic framework, which means that an Earth observer would also conclude that both quantities are conserved.

Question 20

A passenger is on a spaceship traveling at 0.8 c. The passenger observes that her watch
A) is running slow.
B) is running fast.
C) is keeping time as usual.
D) is running slow if the ship is traveling in the positive direction.
E) is running slow if the ship is traveling in the negative direction.

Accepted Answer

The passenger's watch will keep time as usual from her own perspective because time dilation affects observations from an outside frame, not the passenger's own frame of reference.

Quiz 26: Relativity

Protons are sent at 0.900 c down a tube of length 27.0 m. How long does it take for the protons to travel this distance as measured in the lab?

Which of the following is a postulate of special relativity?

A pion has a mean lifetime of 25 ns in its own rest frame. A beam of pions traveling through the laboratory at a speed of 0.60 c is measured to have a mean lifetime of

What are the minimum and maximum values for the Lorentz factor γ, respectively?

Relative to a stationary observer, a moving clock

A conclusion from special relativity is that events that are simultaneous in one inertial frame

At what speed would an astronaut need to travel so that he would age 4.20 years while traveling a distance (as measured by an Earth observer) of 4.20 ly?

Which best describes the proper time interval between two events?

The conclusion of the Michelson-Morley experiments was that

Radioactive particles moving at 0.80 c are measured to have a half-life of 4.3 × 10-8 s. What is their half-life when at rest?

What is the Lorentz factor γ for a speed of 0.800 c?

At what speed would a clock be moving if time dilation caused it to run slow by 50%?

The mean lifetime of muons is 2.2 μs. How fast are they moving if their measured mean lifetime is 1.1 μs?

According to special relativity, what is the speed of light measured in a spaceship traveling at 0.40 c?

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 long does an observer on Earth find that it takes for the spaceship to reach the star?

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 much time passes on the spaceship while traveling to the star?

The principle of relativity states that

Which of the following is the best definition of inertial frame?

An astronaut in a rocket moving with a speed v = 0.6c relative to Earth performs a collision experiment with two small steel balls and concludes the both momentum and energy are conserved. What would an Earth observer conclude?

A passenger is on a spaceship traveling at 0.8 c. The passenger observes that her watch

Radioactive particles moving at 0.80 c are measured to have a half-life of 4.3 × 10^-8 s. What is their half-life when at rest?