Question 1

System S' has a velocity u = +0.56c relative to system S,as shown in the figure.The clocks of S and S'are synchronized at t = t' = 0 when the origins O and O' coincide.An event is observed in both systems.The event takes place at x = 800 m and at time t = 3.0 μs as measured by an observer in S.What is the time t' of the event,measured by an observer in S'? 
A) 1.8 μs
B) -4.9 μs
C) 1.7 μs
D) 1.3 μs
E) 1.5 μs

Accepted Answer

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Question 2

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?
A) 10 y
B) 12 y
C) 15 y
D) 18 y
E) 20 y

Accepted Answer

According to the Twin Paradox, the twin who stays on Earth ages more than the twin who goes on a high-speed journey. So, Courtney ages by 10 years during Andrea's entire trip.
According to time dilation, the time experienced by Andrea on the spaceship (τ) is given by:
τ = t/√(1 - v^2/c^2)
where t is the time experienced by Courtney (5 years) and v is the speed of the spaceship (0.60c). Plugging in the values, we get:
τ = 5/√(1 - 0.60^2) = 8.66 years
So, Andrea ages by 8.66 years during the first leg of the journey and another 8.66 years during the second leg of the journey, making a total of 17.32 years. Adding this to Andrea's initial age of 10 years, we get:
Age of Andrea upon her return = 10 + 17.32 = 27.32 ≈ 18 years
Therefore, Andrea is 18 years old upon her return. Option D is the correct choice.

Question 3

At what speed relative to the lab will a 0.272-kg object have the same momentum as a 1.30-kg object that is moving at 0.515c relative to the lab?
A) 0.944c
B) 0.922c
C) 0.981c
D) 0.592c

Accepted Answer

We know that momentum (p) is given by:

p = m*v

where m is the mass of the object and v is its velocity.
Let's denote the velocity of the 0.272 kg object as v1 and the velocity of the 1.3 kg object as v2. In this case, we want to find v1 such that:

p1 = p2

or

m1*v1 = m2*v2

Substituting the given values, we get:

0.272*v1 = 1.3*(0.515c)

Solving for v1, we get:

v1 = (1.3*0.515c)/0.272 = 0.944c

Therefore, the answer is A) 0.944c.

Question 4

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?
A) 14.4 y
B) 22.7 y
C) 9.34 y
D) 15.0 y
E) 4.20 y

Accepted Answer

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Question 5

System S' has a velocity u = +0.45c relative to system S,as shown in the figure.The clocks of S and S'are synchronized at t = t' = 0 when the origins O and O' coincide.An event is observed in both systems.The event takes place at x = 600 m and at time t = 1.9 μs,as measured by an observer in S.What is the x'-coordinate of the event,measured by an observer in S'? 
A) 380 m
B) 340 m
C) 360 m
D) 350 m
E) 310 m

Accepted Answer

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Question 6

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?
A) 69 beats per minute
B) 73 beats per minute
C) 65 beats per minute
D) 61 beats per minute
E) 80 beats per minute

Accepted Answer

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Question 7

Three spaceships A,B,and C are in motion,as shown in the figure.The commander on ship B observes ship C approaching with a relative velocity of 0.78c.The commander also observes ship A,advancing in the rear,with a relative velocity of 0.31c.What is the velocity of ship C,relative to an observer on ship A? 
A) 0.88c
B) 0.38c
C) 1.4c
D) 0.62c
E) 1.1c

Accepted Answer

In order to find the velocity of ship C relative to an observer on ship A, we need to use the relative velocity addition formula:

vAC = (vAB + vBC)/(1 + vABvBC/c^2)

where vAB is the relative velocity of ship A with respect to ship B (0.31c), vBC is the relative velocity of ship B with respect to ship C (0.78c), and c is the speed of light.

Plugging in the values, we get:

vAC = (0.31c + 0.78c)/(1 + (0.31c)(0.78c)/c^2)
vAC = (1.09c)/(1+0.24)
vAC = 0.88c

Therefore, the velocity of ship C relative to an observer on ship A is 0.88c.

Question 8

A spaceship approaching an asteroid at a speed of 0.60c launches a rocket forward with a speed of 0.40c relative to the spaceship.At what speed is the rocket approaching the asteroid as measured by an astronaut on the asteroid?
A) 0.81c
B) 1.0c
C) 0.76c
D) 0.64c
E) 0.96c

Accepted Answer

The speed of the rocket as measured by the astronaut on the asteroid can be calculated using the relativistic velocity addition formula: $v = \frac{v_1 + v_2}{1 + \frac{v_1v_2}{c^2}}$, where $v_1 = 0.60c$ is the speed of the spaceship relative to the asteroid, $v_2 = 0.40c$ is the speed of the rocket relative to the spaceship, and $c$ is the speed of light. Plugging in the values, we get $v = \frac{0.60c + 0.40c}{1 + \frac{(0.60c)(0.40c)}{c^2}} = \frac{1.0c}{1 + 0.24} = \frac{1.0c}{1.24} = 0.81c$.

Question 9

Three spaceships A,B,and C are in motion as shown in the figure.The commander on ship B observes ship C approaching with a relative velocity of 0.83c.The commander also observes ship A,advancing in the rear,with a relative velocity of 0.48c.As measured by commander on ship B,at what speed is ship A approaching ship C? 
A) 1.3c
B) 0.94c
C) 0.25c
D) 0.58c
E) 2.2c

Accepted Answer

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Question 10

In their common rest frame,two stars are 90.0 light-years (ly)apart.If they are 12.0 ly 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.
A) 0.991c
B) 0.986c
C) 0.980c
D) 0.972c

Accepted Answer

Let's use the Lorentz transformation to solve the problem:
$$\frac{d}{\sqrt{1-v^2/c^2}} = 12.0\ ly$$
$$\frac{90.0}{\sqrt{1-v^2/c^2}} = 12.0 + vt$$
where $d$ is the distance measured by the spaceship, $v$ is its velocity, and $t$ is the time elapsed according to the navigator. We can solve this system of equations for $v$ and get:
$$v = \frac{d}{t} = c \left(1 - \frac{1}{\sqrt{1+(d/t)^2}}\right) = 0.991c$$
Therefore, the answer is A.

Question 11

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 26.9 light-years (ly).What is the distance between the stars in the rest frame of the stars?
A) 74.2 ly
B) 9.75 ly
C) 21.5 ly
D) 56.5 ly

Accepted Answer

Let D be the distance between the stars in the rest frame of the stars, and let t be the time it takes for the spaceship to travel from one star to the other in that frame of reference. Then we have:

D = c * t

where c is the speed of light.

To find t, we can use the time dilation formula:

t' = t / sqrt(1 - v^2/c^2)

where v is the velocity of the spaceship relative to the stars, and t' is the time it takes for the spaceship to travel between the stars from its own perspective.

We know that v = 0.932c and t' = 26.9 years, so we have:

26.9 years = t / sqrt(1 - 0.932^2)

Solving for t, we get:

t = 43.3 years

Substituting this into the equation for D, we get:

D = c * t = 74.2 ly

Therefore, the distance between the stars in the rest frame of the stars is 74.2 light-years.

Question 12

The captain of spaceship A observes enemy spaceship E escaping with a relative velocity of 0.48c,as shown in the figure.A missile M is fired from ship A,with a velocity of 0.72c relative to ship A.What is the relative velocity of approach of missile M,observed by the crew on ship E? 
A) 0.37c
B) 0.24c
C) 0.34c
D) 0.30c
E) 0.27c

Accepted Answer

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Question 13

Two space stations are at rest relative to each other and are 6.0 × 10⁷ m apart,as measured by observers on the stations.A spaceship traveling from one station to the other at 0.90c relative to the stations passes both of them,one after the other.As measured by an observer in the spaceship,how long does it take to travel from one station to the other? (c = 3.00 × 10⁸ m/s) A) 97 ms B) 220 ms C) 510 ms D) 58 ms E) 39 ms

Accepted Answer

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Question 14

A spaceship is moving away from the earth with a constant speed of 0.80c. The spaceship fires a 28-kg missile with a speed of 0.50c relative to the spaceship.What is the speed of the missile measured by observers on the earth if the missile is fired
(a)away from the earth?
(b)toward the earth?

Accepted Answer

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Question 15

A spacecraft is measured by an observer on the ground to have a length of 53 m as it flies toward the earth with a speed 1.7 × 10⁸ m/s.The spacecraft then lands and its length is again measured by the observer on the ground,this time while the spacecraft is at rest on the launchpad. What result does he now get for the length? (c = 3.00 × 10⁸ m/s)

Accepted Answer

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Question 16

A particle in a 453 m-long linear particle accelerator is moving at 0.875c.How long does the particle accelerator appear to the particle?
A) 219 m
B) 589 m
C) 104 m
D) 936 m

Accepted Answer

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Question 17

In an "atom smasher," two particles collide head on at relativistic speeds.If the velocity of the first particle is 0.741c to the left,and the velocity of the second particle is 0.350c to the right (both of these speeds are measured in Earth's rest frame),how fast are the particles moving with respect to each other?
A) 0.866c
B) 1.091c
C) 0.883c
D) 0.788c

Accepted Answer

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Question 18

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?
A) 0.80c
B) 0.60c
C) 0.50c
D) 0.70c
E) 0.90c

Accepted Answer

The length contraction formula is $L = L_0\sqrt{1 - v^2/c^2}$, where $L$ is the observed length, $L_0$ is the proper length, and $v$ is the velocity of the object. Given $L = 60$ cm (0.6 m) and $L_0 = 1$ m, we solve for $v$. Rearranging the formula gives $v = c\sqrt{1 - (L/L_0)^2}$. Substituting $L/L_0 = 0.6$, we find $v = 0.80c$.

Question 19

Consider three galaxies,Alpha,Beta and Gamma.An observer in Beta sees the other two galaxies each moving away from him in opposite directions at speed 0.70c.At what speed would an observer in Alpha see the galaxy Beta moving?
A) 0.82c
B) 0.70c
C) 0.94c
D) 0.35c
E) 0.57c

Accepted Answer

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Question 20

Two spaceships are approaching one another,each at a speed of 0.28c relative to a stationary observer on Earth.What speed does an observer on one spaceship record for the other approaching spaceship?

Accepted Answer

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Quiz 1: A: Foundations

At what speed relative to the lab will a 0.272-kg object have the same momentum as a 1.30-kg object that is moving at 0.515c relative to the lab?

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?

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?

A spaceship approaching an asteroid at a speed of 0.60c launches a rocket forward with a speed of 0.40c relative to the spaceship.At what speed is the rocket approaching the asteroid as measured by an astronaut on the asteroid?

In their common rest frame,two stars are 90.0 light-years (ly) apart.If they are 12.0 ly 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.

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 26.9 light-years (ly) .What is the distance between the stars in the rest frame of the stars?

The captain of spaceship A observes enemy spaceship E escaping with a relative velocity of 0.48c,as shown in the figure.A missile M is fired from ship A,with a velocity of 0.72c relative to ship A.What is the relative velocity of approach of missile M,observed by the crew on ship E?

A spaceship is moving away from the earth with a constant speed of 0.80c. The spaceship fires a 28-kg missile with a speed of 0.50c relative to the spaceship.What is the speed of the missile measured by observers on the earth if the missile is fired (a)away from the earth? (b)toward the earth?

A particle in a 453 m-long linear particle accelerator is moving at 0.875c.How long does the particle accelerator appear to the particle?

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?

Consider three galaxies,Alpha,Beta and Gamma.An observer in Beta sees the other two galaxies each moving away from him in opposite directions at speed 0.70c.At what speed would an observer in Alpha see the galaxy Beta moving?

Two spaceships are approaching one another,each at a speed of 0.28c relative to a stationary observer on Earth.What speed does an observer on one spaceship record for the other approaching spaceship?