Question 1

One 20-year-old twin brother takes a space trip with a speed of 0.80c for 30 years according to a clock on his spaceship. At the end of this trip, what are (a)his own age and (b)the age of his Earth-based twin brother?
A)(a)20 y (b)30 y
B)(a)30 y (b)50 y
C)(a)50 y (b)70 y
D)(a)70 y (b)90 y

Accepted Answer

According to the theory of relativity, time dilation occurs at high speeds. This means that time moves slower for the twin traveling at 0.80c compared to the twin on Earth. 
Using the formula for time dilation, we can find the time experienced by the traveling twin (t') as:
t' = t / sqrt(1 - v^2/c^2)
where t is the time experienced by the Earth-based twin, v is the velocity of the traveling twin, and c is the speed of light. 
Plugging in the values, we get:
t' = 30 / sqrt(1 - 0.80^2) = 75 years
So, the traveling twin has aged 75 - 30 = 45 years during the trip. 
Therefore, at the end of the trip, 
(a) the traveling twin's age = 20 + 45 = 65 years
(b) the Earth-based twin's age = 20 + 30 = 50 years 
The closest option to these values is option C: (a)50 y (b)70 y.

Question 2

A spaceship with a constant velocity of 0.800c relative to Earth travels to the star that is 4.30 light-years from Earth. What distance does the space ship travel as measured by a passenger on the ship?
A)2.58 light-years
B)3.52 light-years
C)4.12 light-years
D)4.30 light-years

Accepted Answer

According to the theory of relativity, distances appear to be shortened in the direction of motion when observed from a frame of reference in motion relative to an observer. This phenomenon is known as length contraction.

The distance between Earth and the star, as measured by an observer on Earth, is 4.30 light-years. However, since the spaceship is moving at a velocity of 0.800c relative to Earth, its length in the direction of motion will appear to be contracted.

Using the formula for length contraction, the length of the spaceship as measured by an observer on Earth will be:

L' = L * sqrt(1 - v^2/c^2)

where L is the rest length of the spaceship, v is its velocity relative to the observer, c is the speed of light, and L' is the contracted length.

Plugging in the values, we get:

L' = (4.30 ly) * sqrt(1 - 0.800^2) = 2.58 ly

Therefore, the distance the spaceship travels as measured by a passenger on the ship will also be 2.58 light-years, since the passenger is at rest relative to the spaceship.

Question 3

Astronaut Hans leaves Earth in a spaceship traveling at a speed of 0.280c relative to the observer Brian at rest on Earth. Hans is holding a pencil at an angle of 30° with the direction of travel, as seen by Hans. What angle does the pencil make with the direction of travel, as seen by Brian on Earth?
A)29°
B)30°
C)31°
D)33°
E)90°

Accepted Answer

The answer of Astronaut Hans leaves Earth in a spaceship...

Question 4

A rocket is traveling away from Earth at 90% the speed of light. The captain turns on a laser light. What is the speed of the light as measured by an Earth-based astronomer if the captain aims the laser (a)directly forward or (b)directly backward?

Accepted Answer

The answer of A rocket is traveling away from Earth...

Question 5

A set of twins, Andrea and Courtney, are initially 10 years old. While Courtney remains on Earth, Andrea rides on a spaceship that travels away from Earth at a speed of 0.60c for 10 years (as measured by Courtney). At the end of the trip, Courtney is 20 years old. How old is Andrea?
A)10 years
B)12 years
C)18 years
D)20 years

Accepted Answer

According to special relativity, time dilation occurs for an object in motion relative to an observer. Therefore, Andrea experiences less time compared to Courtney. Using the time dilation formula (t'=t/sqrt(1-v^2/c^2)), we can calculate the time experienced by Andrea: 
t' = 10/sqrt(1-0.60^2)
t' = 18.75 years
Therefore, Andrea is 18.75 years old at the end of the trip.

Question 6

Two stars are at rest relative to each other. A spaceship is moving between these two stars at 0.932c relative to each star. To someone in the ship, the distance between the two stars appears to be $26.4 \text { light-years. }$ What is the distance between the stars in their own rest frame?
A)72.8 light-years
B)9.57 light-years
C)21.1 light-years
D)55.4 light-years

Accepted Answer

The distance in the rest frame is longer due to length contraction. Using the Lorentz factor for 0.932c, the rest distance is 26.4 light-years divided by the Lorentz factor, which gives approximately 72.8 light-years.

Question 7

The figure shows a right-angled construction frame ABC. When measured at rest, it has dimensions as follows: AB = 18.9 m, AC = 10.0 m, and BC = 16.0 m. This frame is now given a new velocity of 0.420c, relative to Earth, in a direction parallel to AC. How long is part BC of this frame with the new velocity as measured by Earth-based engineers? 
A)16.0 m
B)14.0 m
C)12.0 m
D)10.0 m
E)18.0 m

Accepted Answer

The answer of The figure shows a right-angled construction frame...

Question 8

A spacecraft is measured by a scientist on the ground to have a length of 81 m as it flies overhead with a speed $2.4 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s }$ The spacecraft then lands and its length is again measured by the scientist at rest on the ground. What result does he now get for the length of the spacecraft? (c = 3.0 × 10⁸ m/s)

Accepted Answer

The answer of A spacecraft is measured by a scientist...

Question 9

How fast must a meter stick move, relative to the frame of the observer making the measurement, so its length will be measured to be 0.60 m?
A)0.40c
B)0.60c
C)0.70c
D)0.80c
E)0.90c

Accepted Answer

The length contraction formula is $L = L_0\sqrt{1 - \frac{v^2}{c^2}}$, where $L$ is the contracted length, $L_0$ is the rest length, $v$ is the velocity of the object, and $c$ is the speed of light. Rearranging for $v$, given $L_0 = 1$ m and $L = 0.60$ m, we find $v = c\sqrt{1 - \left(\frac{L}{L_0}\right)^2} = c\sqrt{1 - 0.36} = 0.8c$.

Question 10

Astronaut Sarah leaves Earth in a spaceship at a speed of 0.280c relative to Earth. Sarah's destination is a star system 12.5 light-years away (one light-year is the distance light travels in one year). According to Sarah, how far did she travel during the trip?
A)13.0 light-years
B)12.5 light-years
C)12.0 light-years
D)11.5 light-years
E)11.0 light-years

Accepted Answer

According to special relativity, distances and times are different for observers moving relative to each other. From Sarah's perspective, she is stationary in the spaceship and the star system is moving towards her at a speed of 0.280c. Therefore, she measures a contracted distance to the star system:
length = length0 / γ
where length0 is the length of the star system in its rest frame (as measured by an observer at rest relative to it) and γ is the Lorentz factor given by:
γ = 1 / sqrt(1 - (v/c)^2)
where v is the relative speed of the two frames (0.280c in this case) and c is the speed of light. Plugging in the values, we get:
γ = 1 / sqrt(1 - (0.280)^2) ≈ 1.0564
length = 12.5 light-years / 1.0564 ≈ 11.83 light-years
However, we also need to account for time dilation, which means that Sarah measures a longer time for the trip than an observer at rest on Earth. The time it takes for Sarah to travel to the star system is given by:
time = length / v
where v is her speed relative to Earth (0.280c + c = 1.280c). Plugging in the values, we get:
time = 11.83 light-years / (0.280c + c) ≈ 9.218 years
According to the principle of relativity, the distance and time measurements made by Sarah are equally valid as those made by an observer on Earth. Therefore, we can use the distance and time measurements made on Earth to calculate the distance that Sarah traveled:
distance = speed × time = c × 9.218 years ≈ 9.218 light-years
Therefore, the total distance traveled by Sarah, as measured by an observer on Earth, is the sum of the contracted distance to the star system and the distance traveled in the Earth's rest frame:
total distance = length + distance = 11.83 light-years + 9.218 light-years ≈ 21.048 light-years
However, we need to convert this to the distance that Sarah herself measured during the trip, which is given by:
distance' = total distance / γ ≈ 19.944 light-years
Therefore, according to Sarah, she traveled a distance of approximately 19.944 light-years during the trip, which is closest to choice C) 12.0 light-years (the contracted distance to the star system).

Question 11

A person in a rocket ship traveling past the earth at a speed of 0.500c fires a laser gun in the forward direction. With what speed does an observer on Earth measure for the light pulse?

Accepted Answer

The answer of A person in a rocket ship traveling...

Question 12

How fast would a rocket ship have to move past Earth to contract to half of its proper length as observed by an Earth-based physicist?

Accepted Answer

The answer of How fast would a rocket ship have...

Question 13

From the Earth, the Earthlings see the Enterprise approaching from the west at 0.800c and the Klingons approaching from the east at 0.900c. What speed, in terms of the speed of light, does the crew of the Enterprise measure for the Klingon ship?

Accepted Answer

The answer of From the Earth, the Earthlings see the...

Question 14

The figure shows a right-angled construction frame ABC. When measured at rest, it has dimensions as follows: AB = 26.0 m, AC = 10.0 m, and BC = 24.0 m. This frame is now given a new velocity of 0.760c, relative to Earth, in a direction parallel to AC. How long is part AC of this frame with the new velocity as measured by Earth-based engineers? 
A)6.50 m
B)25.4 m
C)24.3 m
D)23.7 m
E)24.9 m

Accepted Answer

The answer of The figure shows a right-angled construction frame...

Question 15

A stretch of land is 12.5 km long on a map.
(a)What would be its length to an observer moving by, parallel to the stretch, at $0.9500 c ?$ (b)Which is the proper length for this stretch of land?
(c)How fast relative to Earth would the observer have to move for the land to appear 6.50 km long? Express your answer in terms of the speed of light.

Accepted Answer

The answer of A stretch of land is 12.5 km...

Question 16

Two spaceships are approaching one another, each at a speed of 0.31c relative to a stationary observer on Earth. What speed, in terms of the speed of light, does an observer on one spaceship record for the other approaching spaceship?

Accepted Answer

The answer of Two spaceships are approaching one another, each...

Question 17

Astronaut Jill leaves Earth in a spaceship and is now traveling at a speed of 0.280c relative to an observer on Earth. When Jill left Earth, the spaceship was equipped with all kinds of scientific instruments, including a meter stick. Now that Jill is underway, how long does she measure the meter stick to be?
A)1.00 m
B)0.960 m
C)1.04 m
D)0.280 m
E)1.28 m

Accepted Answer

Jill, being in the same inertial frame as the meter stick, would measure it to be its proper length, which is 1.00 m. The effects of length contraction are observed from a different inertial frame, not within the same frame as the object.

Question 18

A particle in a 799=m-long linear particle accelerator is moving at $0.875 c$ How long does the particle accelerator appear to the particle?
A)387 m
B)1039 m
C)184 m
D)1651 m

Accepted Answer

The length contraction formula is $ L = L_0 \sqrt{1 - v^2/c^2} $. Here, $ L_0 = 799 \, \text{m} $ and $ v = 0.875c $. Plugging in the values, $ L = 799 \sqrt{1 - (0.875)^2} \approx 387 \, \text{m} $.

Question 19

A starship is constructed to be 100 m long in the factory. It is launched and as it coasts by Earth, it is measured to be shortened to 5.48 m long.
(a)How fast, in terms of the speed of light, is it moving past Earth?
(b)What length does the ship's crew measure for the ship?

Accepted Answer

The answer of A starship is constructed to be 100...

Question 20

Two stars are at rest relative to each other. In their common rest frame, these stars are 90.0 light-years apart. If they are measured to be $68.7 \text { light-years }$ apart by the navigator of a spaceship traveling between them, how fast is the spaceship moving relative to these stars?
A)0.646c
B)0.604c
C)0.558c
D)0.505c

Accepted Answer

The length contraction formula is $ L = L_0 \sqrt{1 - v^2/c^2} $, where $ L_0 = 90.0 $ light-years and $ L = 68.7 $ light-years. Solving for $ v $, we find $ v \approx 0.646c $.

Quiz 27: Relativity

One 20-year-old twin brother takes a space trip with a speed of 0.80c for 30 years according to a clock on his spaceship. At the end of this trip, what are (a) his own age and (b) the age of his Earth-based twin brother?

A spaceship with a constant velocity of 0.800c relative to Earth travels to the star that is 4.30 light-years from Earth. What distance does the space ship travel as measured by a passenger on the ship?

A rocket is traveling away from Earth at 90% the speed of light. The captain turns on a laser light. What is the speed of the light as measured by an Earth-based astronomer if the captain aims the laser (a)directly forward or (b)directly backward?

A set of twins, Andrea and Courtney, are initially 10 years old. While Courtney remains on Earth, Andrea rides on a spaceship that travels away from Earth at a speed of 0.60c for 10 years (as measured by Courtney) . At the end of the trip, Courtney is 20 years old. How old is Andrea?

Two stars are at rest relative to each other. A spaceship is moving between these two stars at 0.932c relative to each star. To someone in the ship, the distance between the two stars appears to be $26.4 \text { light-years. }$ What is the distance between the stars in their own rest frame?

How fast must a meter stick move, relative to the frame of the observer making the measurement, so its length will be measured to be 0.60 m?

Astronaut Sarah leaves Earth in a spaceship at a speed of 0.280c relative to Earth. Sarah's destination is a star system 12.5 light-years away (one light-year is the distance light travels in one year) . According to Sarah, how far did she travel during the trip?

A person in a rocket ship traveling past the earth at a speed of 0.500c fires a laser gun in the forward direction. With what speed does an observer on Earth measure for the light pulse?

How fast would a rocket ship have to move past Earth to contract to half of its proper length as observed by an Earth-based physicist?

From the Earth, the Earthlings see the Enterprise approaching from the west at 0.800c and the Klingons approaching from the east at 0.900c. What speed, in terms of the speed of light, does the crew of the Enterprise measure for the Klingon ship?

Two spaceships are approaching one another, each at a speed of 0.31c relative to a stationary observer on Earth. What speed, in terms of the speed of light, does an observer on one spaceship record for the other approaching spaceship?

A particle in a 799=m-long linear particle accelerator is moving at $0.875 c$ How long does the particle accelerator appear to the particle?

A starship is constructed to be 100 m long in the factory. It is launched and as it coasts by Earth, it is measured to be shortened to 5.48 m long. (a)How fast, in terms of the speed of light, is it moving past Earth? (b)What length does the ship's crew measure for the ship?

Two stars are at rest relative to each other. In their common rest frame, these stars are 90.0 light-years apart. If they are measured to be $68.7 \text { light-years }$ apart by the navigator of a spaceship traveling between them, how fast is the spaceship moving relative to these stars?

Quiz 27: Relativity

One 20-year-old twin brother takes a space trip with a speed of 0.80c for 30 years according to a clock on his spaceship. At the end of this trip, what are (a) his own age and (b) the age of his Earth-based twin brother?

A spaceship with a constant velocity of 0.800c relative to Earth travels to the star that is 4.30 light-years from Earth. What distance does the space ship travel as measured by a passenger on the ship?

A rocket is traveling away from Earth at 90% the speed of light. The captain turns on a laser light. What is the speed of the light as measured by an Earth-based astronomer if the captain aims the laser (a)directly forward or (b)directly backward?

A set of twins, Andrea and Courtney, are initially 10 years old. While Courtney remains on Earth, Andrea rides on a spaceship that travels away from Earth at a speed of 0.60c for 10 years (as measured by Courtney) . At the end of the trip, Courtney is 20 years old. How old is Andrea?

How fast must a meter stick move, relative to the frame of the observer making the measurement, so its length will be measured to be 0.60 m?

Astronaut Sarah leaves Earth in a spaceship at a speed of 0.280c relative to Earth. Sarah's destination is a star system 12.5 light-years away (one light-year is the distance light travels in one year) . According to Sarah, how far did she travel during the trip?

A person in a rocket ship traveling past the earth at a speed of 0.500c fires a laser gun in the forward direction. With what speed does an observer on Earth measure for the light pulse?

How fast would a rocket ship have to move past Earth to contract to half of its proper length as observed by an Earth-based physicist?

From the Earth, the Earthlings see the Enterprise approaching from the west at 0.800c and the Klingons approaching from the east at 0.900c. What speed, in terms of the speed of light, does the crew of the Enterprise measure for the Klingon ship?

Two spaceships are approaching one another, each at a speed of 0.31c relative to a stationary observer on Earth. What speed, in terms of the speed of light, does an observer on one spaceship record for the other approaching spaceship?

A particle in a 799=m-long linear particle accelerator is moving at 0.875c0.875 c0.875c How long does the particle accelerator appear to the particle?

A starship is constructed to be 100 m long in the factory. It is launched and as it coasts by Earth, it is measured to be shortened to 5.48 m long. (a)How fast, in terms of the speed of light, is it moving past Earth? (b)What length does the ship's crew measure for the ship?

A particle in a 799=m-long linear particle accelerator is moving at $0.875 c$ How long does the particle accelerator appear to the particle?