Question 1

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?

Accepted Answer

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

Question 2

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?

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 = 0.646c $.

Question 3

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?

Accepted Answer

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

Question 4

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 $0.9500 \mathrm { c } ?$ (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 5

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 6

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?

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Question 7

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?

Accepted Answer

The distance in the rest frame is longer due to length contraction. Using the formula $L = L_0 \sqrt{1 - v^2/c^2}$, where $L$ is the contracted length (26.4 light-years), $v = 0.932c$, and solving for $L_0$, we find $L_0 = 72.8$ light-years.

Question 8

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?

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 9

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?

Accepted Answer

According to the theory of relativity, time dilation occurs at high speeds. Therefore, while Andrea is traveling on the spaceship, time is moving slower for her than for Courtney on Earth. Using the time dilation equation (t' = t / sqrt(1 - v^2/c^2)), we can calculate that Andrea experiences a time of approximately 8.66 years (since v = 0.60c and t = 10 years). Therefore, Andrea is 10 + 8.66 = 18.66 years old at the end of the trip, which we can round to 18 years.

Question 10

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?

Accepted Answer

The distance measured by a passenger on the ship is shorter due to length contraction, a phenomenon of special relativity. The formula for length contraction is $L = L_0\sqrt{1 - v^2/c^2}$, where $L_0$ is the rest length (4.30 light-years), $v$ is the velocity of the ship (0.800c), and $c$ is the speed of light. Plugging in the numbers, $L = 4.30\sqrt{1 - (0.800)^2} = 4.30\sqrt{1 - 0.64} = 4.30\sqrt{0.36} = 4.30 \times 0.6 = 2.58$ light-years.

Question 11

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 12

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 13

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?

Accepted Answer

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

Question 14

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 15

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?

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 16

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 17

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?

Accepted Answer

We can use the formula for length contraction:
$L_{s}=L_{o}\sqrt{1-\frac{v^{2}}{c^{2}}}$
where $L_{s}$ is the contracted length, $L_{o}$ is the rest length, $v$ is the velocity of the object, and $c$ is the speed of light.
Plugging in the values, we get:
$L_{s}=799\sqrt{1-(0.875)^{2}}=387$ m
Therefore, the particle accelerator appears to be 387 m long to the particle.

Question 18

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?

Accepted Answer

According to the theory of relativity, time dilation occurs as an object approaches the speed of light. The faster the object moves, the slower time passes for it relative to a stationary observer. In this case, the twin brother traveling at 0.8c experiences time dilation, meaning that time passes slower for him compared to his twin brother on Earth.

Using the formula t' = t/√(1-v^2/c^2), where t' is the time dilation experienced by the traveling twin, t is the time experienced by the Earth-bound twin, v is the velocity of the traveling twin, and c is the speed of light, we can calculate the age difference between the two twin brothers.

Plugging in v = 0.8c, t = 30 years, and t' = ? into the formula, we get:

t' = t/√(1-v^2/c^2)
t' = 30/√(1-0.8^2)
t' = 30/√(0.36)
t' = 30/0.6
t' = 50 years

Therefore, the traveling twin brother will age 50 years while his Earth-based twin brother will age 70 years (20 years older than the traveling twin brother). The best choice is C.

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

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...

Quiz 27: Relativity

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?

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 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?

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 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 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?

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 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 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?

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?

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?

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 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?

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?

Quiz 27: Relativity

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 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?

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?

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 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 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?

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?

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?

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 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?

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?

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?