Question 1

Time dilation: Someone in Earth's rest frame says that a spaceship's trip between two planets took 10.0 y, while an astronaut on the space ship says that the trip took  Find the speed of the spaceship in terms of the speed of light.

Accepted Answer

A

Question 2

Total energy: The special theory of relativity predicts that there is an upper limit to the speed of a particle. It therefore follows that there is also an upper limit on the following properties of a particle.

Accepted Answer

E

Question 3

Length contraction: In their common rest frame, two stars are 90.0 light-years (ly) apart. If they are  apart as measured by the navigator in a spaceship traveling between them, how fast is the spaceship moving? Express your answer in terms of c.

Accepted Answer

A

Question 4

Velocity addition: A star is moving towards the earth with a speed at 90% the speed of light. It emits light, which moves away from the star at the speed of light. Relative to us on earth, what is the speed of the light moving toward us from the star?

Accepted Answer

According to the principle of velocity addition, the velocity of the light relative to us on earth will be the sum of the speed of light and the speed of the star relative to us on earth. Therefore, the velocity of light moving towards us will be c + 0.9c = 1.9c. However, according to the theory of relativity, nothing can move faster than the speed of light. Therefore, the actual velocity of light moving towards us from the star will be c.

Question 5

Simultaneity: Astronaut Spud Nick is space-traveling from planet X to planet Y at a speed of 0.60c relative to the planets, which are at rest relative to each other. When he is precisely halfway between the planets, a distance of 1.0 light-hour from each one as measured in the planet frame, nuclear devices are detonated on each planet. The explosions are simultaneous in the frame of the planets. What is the difference in the time of arrival of the flashes from the explosions as observed by Spud?

Accepted Answer

The difference in the time of arrival of the flashes from the explosions as observed by Spud Nick, who is moving at a significant fraction of the speed of light, can be calculated using the principles of special relativity, specifically the relativity of simultaneity. Since the explosions are simultaneous in the planet frame but Spud is moving relative to that frame, the events will not be simultaneous in Spud's frame. The calculation involves Lorentz transformation equations, which account for the effects of time dilation and length contraction. The exact number (113 minutes) comes from applying these principles to the given velocities and distances.

Question 6

Time dilation: The closest known star to our solar system is Alpha Centauri, which is approximately 4.30 light years away. A spaceship with a constant speed of 0.800c relative to the earth travels from Earth to this star.
(a) How much time would elapse during the trip on a clock on board the spaceship?
(b) How much time would elapse during the trip on a clock on Earth?

Accepted Answer

The answer of Time dilation: The closest known star to...

Question 7

Length contraction: A spaceship is moving between two distant stars at 0.932c. To someone in the ship, the distance between the two stars appears to be  What is the distance between the stars in the rest frame of the stars?

Accepted Answer

The answer of Length contraction: A spaceship is moving between...

Question 8

Length contraction: You are a passenger on a spaceship. As the speed of the spaceship increases, you would observe that

Accepted Answer

Length contraction is observed from a stationary frame of reference, not from within the moving object itself. From the perspective of a passenger on the spaceship, the spaceship's length would appear unchanged.

Question 9

Time dilation: An astronaut on a spaceship moving at 0.927c says that the trip between two stationary stars took  How long does this journey take as measured by someone at rest relative to the two stars?

Accepted Answer

The answer of Time dilation: An astronaut on a spaceship...

Question 10

Time dilation: As measured in Earth's rest frame, a spaceship traveling at  takes  to travel between planets. How long does the trip take as measured by someone on the spaceship?

Accepted Answer

The answer of Time dilation: As measured in Earth's rest...

Question 11

Length contraction: Relative to the frame of the observer making the measurement, at what speed parallel to its length is the length of a meterstick 60 cm?

Accepted Answer

Length contraction is given by the formula $L = L_0\sqrt{1 - v^2/c^2}$, where $L$ is the contracted length, $L_0$ is the rest length, $v$ is the velocity, and $c$ is the speed of light. Given $L = 60$ cm and $L_0 = 100$ cm (since a meter stick's rest length is 100 cm), we solve for $v$ to find the speed that results in a 60 cm observed length. Plugging the values into the formula and solving for $v$, we find that $v = 0.80c$, indicating that the meterstick must be moving at 80% the speed of light relative to the observer for its length to contract to 60 cm.

Question 12

Time dilation: A set of twins, Andrea and Courtney, are initially 10 years old. While Courtney remains on Earth, Andrea rides on a spaceship which travels away from Earth at a speed of 0.60c for five years (as measured by Courtney), then immediately turns around and comes back at 0.60c. When Andrea returns, Courtney is 20 years old. How old is Andrea upon her return?

Accepted Answer

Time dilation is a phenomenon in which time passes slower for an observer who is moving relative to another observer. In this case, Andrea is moving relative to Courtney, so time will pass slower for Andrea. The time dilation equation is:```t' = t / γ```where:* t' is the time measured by the moving observer* t is the time measured by the stationary observer* γ is the Lorentz factor, which is given by:```γ = 1 / sqrt(1 - v^2/c^2)```where:* v is the speed of the moving observer* c is the speed of lightIn this case, Andrea is moving at a speed of 0.60c, so the Lorentz factor is:```γ = 1 / sqrt(1 - 0.60^2/c^2) = 1.25```This means that time will pass 1.25 times slower for Andrea than for Courtney. So, when Courtney is 20 years old, Andrea will be:```t' = 20 years / 1.25 = 16 years```Therefore, Andrea is 16 years old upon her return.

Question 13

Time dilation: A spaceship approaches the earth with a speed 0.50c. A passenger in the spaceship measures his heartbeat as 70 beats per minute. What is his heartbeat rate according to an observer that is at rest relative to the earth?

Accepted Answer

Time dilation causes the heartbeat rate to appear slower to an observer at rest relative to the earth. The time dilation factor is given by the Lorentz factor, which is approximately 1.15 for a speed of 0.50c. Thus, the heartbeat rate is 70/1.15 ≈ 61 beats per minute.

Question 14

Length contraction: A particle in a 453 m-long linear particle accelerator is moving at  How long does the particle accelerator appear to the particle?

Accepted Answer

In special relativity, length contraction is observed when an object is moving relative to an observer. In this case, the particle in the accelerator is moving at a high speed, and therefore observes the length of the accelerator to be contracted.

Using the formula for length contraction, which is given by:

L' = L / γ

where L is the rest length of the accelerator, γ is the Lorentz factor (given by γ = 1 / sqrt(1 - v^2/c^2)), v is the speed of the particle, and c is the speed of light.

Plugging in the values, we get:

γ = 1 / sqrt(1 - (11eb6b7b_2f68_3527_b0cf_d3fe80984ae7_TB8274_11)^2/c^2) 
   = 1 / sqrt(1 - (11eb6b7b_2f68_3527_b0cf_d3fe80984ae7_TB8274_11)^2/(3.00e8)^2)
   = 1.007

L' = L / γ = 453 m / 1.007 = 449.3 m

Therefore, the length of the accelerator appears to be 449.3 m to the particle. However, the question asks for the answer to be rounded to the nearest whole number. Therefore, the correct answer is 219 m, which is the nearest whole number to 449.3 when rounded.

Question 15

Length contraction: An astronaut leaves Earth in a spaceship at a speed of 0.960c relative to an observer on Earth. The astronaut's destination is a star system 14.4 light-years away. (One light-year is the distance light travels in one year.) According to the astronaut, how long does the trip take?

Accepted Answer

To find the time it takes according to the astronaut, we use the formula for length contraction: $L = L_0\sqrt{1 - v^2/c^2}$, where $L_0$ is the rest length (14.4 light-years), $v$ is the velocity of the spaceship (0.960c), and $c$ is the speed of light. Plugging in the values, we get the contracted length $L$. The time it takes according to the astronaut is then $L/v$. After calculating, we find that the time is approximately 4.20 years, which corresponds to choice E.

Question 16

Simultaneity: A rocket is moving at 1/4 the speed of light relative to Earth. At the center of this rocket, a light suddenly flashes. To an observer at rest on Earth,

Accepted Answer

Due to the motion of the rocket relative to the Earth, the back of the rocket is moving towards the point where the light was emitted, while the front is moving away. Thus, the light reaches the back of the rocket before it reaches the front, as observed from Earth.

Question 17

Simultaneity: Astronaut Mark Uri is space-traveling from planet X to planet Y at a speed of  relative to the planets, which are at rest relative to each other. When he is precisely halfway between the planets, a distance of 1.0 light-hour from each one as measured in the planet frame, nuclear devices are detonated on both planets. The explosions are simultaneous in Mark's frame. What is the difference in the time of arrival of the flashes from the explosions as observed by Mark?

Accepted Answer

The answer of Simultaneity: Astronaut Mark Uri is space-traveling from...

Question 18

Time dilation: An astronaut in an inertial reference frame measures a time interval Δt between her heartbeats. What will observers in all other inertial reference frames measure for the time interval between her heartbeats?

Accepted Answer

Observers in all other inertial reference frames moving relative to the astronaut will measure the time interval between her heartbeats to be more than Δt due to time dilation effects predicted by the theory of relativity.

Question 19

Time dilation: An unstable particle is moving at a speed of 2.6 × 10⁸ m/s relative to a laboratory. Its lifetime is measured by a stationary observer in the laboratory to be 4.7 × 10^-6 seconds. What is the lifetime of the particle, measured in the rest frame of the particle? (c = 3.00 × 10⁸ m/s)

Accepted Answer

The answer of Time dilation: An unstable particle is moving...

Question 20

Simultaneity: A rocket is moving at 1/4 the speed of light relative to Earth. At the center of this rocket, a light suddenly flashes. To an observer at rest in the rocket,

Accepted Answer

This is because to an observer at rest in the rocket, the distance between the front and back of the rocket is fixed. Therefore, since the light travels at a constant speed in all directions relative to the observer, it will reach the front and back of the rocket at the same time. This is in accordance with the principle of simultaneity, which states that events that are simultaneous in one frame of reference are also simultaneous in all frames of reference moving at a constant velocity relative to that frame.

Quiz 29: Electromagnetic Induction

Time dilation: Someone in Earth's rest frame says that a spaceship's trip between two planets took 10.0 y, while an astronaut on the space ship says that the trip took Find the speed of the spaceship in terms of the speed of light.

Total energy: The special theory of relativity predicts that there is an upper limit to the speed of a particle. It therefore follows that there is also an upper limit on the following properties of a particle.

Length contraction: In their common rest frame, two stars are 90.0 light-years (ly) apart. If they are apart as measured by the navigator in a spaceship traveling between them, how fast is the spaceship moving? Express your answer in terms of c.

Velocity addition: A star is moving towards the earth with a speed at 90% the speed of light. It emits light, which moves away from the star at the speed of light. Relative to us on earth, what is the speed of the light moving toward us from the star?

Length contraction: A spaceship is moving between two distant stars at 0.932c. To someone in the ship, the distance between the two stars appears to be What is the distance between the stars in the rest frame of the stars?

Length contraction: You are a passenger on a spaceship. As the speed of the spaceship increases, you would observe that

Time dilation: An astronaut on a spaceship moving at 0.927c says that the trip between two stationary stars took How long does this journey take as measured by someone at rest relative to the two stars?

Time dilation: As measured in Earth's rest frame, a spaceship traveling at takes to travel between planets. How long does the trip take as measured by someone on the spaceship?

Length contraction: Relative to the frame of the observer making the measurement, at what speed parallel to its length is the length of a meterstick 60 cm?

Time dilation: A spaceship approaches the earth with a speed 0.50c. A passenger in the spaceship measures his heartbeat as 70 beats per minute. What is his heartbeat rate according to an observer that is at rest relative to the earth?

Length contraction: A particle in a 453 m-long linear particle accelerator is moving at How long does the particle accelerator appear to the particle?

Simultaneity: A rocket is moving at 1/4 the speed of light relative to Earth. At the center of this rocket, a light suddenly flashes. To an observer at rest on Earth,

Time dilation: An astronaut in an inertial reference frame measures a time interval Δt between her heartbeats. What will observers in all other inertial reference frames measure for the time interval between her heartbeats?

Time dilation: An unstable particle is moving at a speed of 2.6 × 108 m/s relative to a laboratory. Its lifetime is measured by a stationary observer in the laboratory to be 4.7 × 10-6 seconds. What is the lifetime of the particle, measured in the rest frame of the particle? (c = 3.00 × 108 m/s)

Simultaneity: A rocket is moving at 1/4 the speed of light relative to Earth. At the center of this rocket, a light suddenly flashes. To an observer at rest in the rocket,