Question 1

At rest, pions have a proper lifetime of 26 ns. What is the speed of a beam of pions if their observed lifetime were measured to be 64 ns ? This is a real experiment that has been performed and found to be consistent with special relativity.)
A) 0.96c
B) 0.84c
C) 0.70c
D) 0.41c
E) 0.91c

Accepted Answer

The answer of At rest, pions have a proper lifetime...

Question 2

A star system is 100 light years away. If a spacecraft travels to the star system at 97% the speed of light, then what is the ratio of the time for the journey measured by an observer left on Earth divided by the time measured by the astronaut?
A) 4.1
B) 17
C) 1.0
D) 1.3
E) 2.4

Accepted Answer

4.1The time dilation effect described by Einstein's theory of relativity explains that time passes at different rates for observers in different inertial frames. For an observer on Earth, the spacecraft's journey appears to take slightly over 100 years because it's traveling close to the speed of light (100/0.97 ≈ 103.1 years). For the astronaut on the spacecraft, however, the journey's duration is significantly shorter due to time dilation. The time dilation factor (gamma) can be calculated using the formula γ = 1 / sqrt(1 - v^2/c^2), where v is the velocity of the spacecraft and c is the speed of light. Plugging in 0.97c for v gives γ ≈ 4.1, meaning the astronaut experiences time at a rate 4.1 times slower than the observer on Earth. Thus, the ratio of the time for the journey measured by an observer on Earth to the time measured by the astronaut is approximately 4.1.

Question 3

Castor and Pollux, identical twins, get into their individual spaceships and blast off simultaneously. Castor's is the faster ship, and both return to Earth simultaneously. Which of the following statements is true?
A) Castor is older only if he traveled in the same direction as Pollux.
B) Castor is older only if he traveled in the opposite direction to Pollux.
C) Pollux is older only if he traveled in the same direction as Castor.
D) Pollux is older only if he traveled in the opposite direction to Castor.
E) Pollux is older.

Accepted Answer

According to the theory of relativity, the twin who travels at a higher speed (Castor) will age more slowly than the twin who travels at a lower speed (Pollux). Therefore, Pollux will be older when they return to Earth.

Question 4

A 26-year-old biologist makes a trip to study alien life-forms on a distant planet 10 light years away. The round-trip including a stay of 1 year in the distant planet takes 21.5 years according to the clock on earth. The biologist's son is 3 years old when she leaves. Assume that the planet is not moving with respect to Earth, and the speed is the same for both the outbound and inbound parts of the journey. What is the difference in age between the biologist and her son after she returns to Earth?
A) 7.0 years
B) 16.8 years
C) 23.0 years
D) 30.0 years
E) 37.0 years

Accepted Answer

The answer of A 26-year-old biologist makes a trip to...

Question 5

Spaceship Solange passes spaceship Hari with a relative velocity of 0.6c. An astronaut aboard Hari observes the length of Solange to be 30.5 m. The astronauts aboard Solange claim that the length of their spaceship is
A) 18.3 m.
B) 24.4 m.
C) 30.5 m.
D) 38.1 m.
E) 39.6 m.

Accepted Answer

The answer of Spaceship Solange passes spaceship Hari with a...

Question 6

A standard Canadian football field is 100 m long and 60 m wide. A UFO streaks from one end of the field to the other at a speed of 0.6c relative to the players on the field. An observer in the UFO measures the length of the field to be
A) 60 m.
B) 80 m.
C) 100 m.
D) 125 m.
E) 167 m.

Accepted Answer

The answer of A standard Canadian football field is 100...

Question 7

A beam of unstable neutral pions is observed to travel on average 8 m before they decay. If their proper lifetime is 0.8 × 10^-8 s, then calculate how fast the beam is traveling. A) 0.96c B) 0.92c C) 0.29c D) 0.98c E) 3.5c

Accepted Answer

The average distance traveled is related to the time dilation effect in special relativity. Using the formula for time dilation, the speed can be calculated as v = d / (γ * τ₀), where γ is the Lorentz factor. Solving for v gives approximately 0.96c.

Question 8

Systems S and S' are in relative motion as indicated in the sketch. Metersticks and clocks in S' agree with metersticks and clocks in S when t = t' = 0. An observer in S' notes that two events occurring at x₁' and x₂' are simultaneous; i.e., Δx' > 0 and Δt' = 0 for the observer in S'. An observer in S, seeing the same two events, may find that A) Δx > Δx' and Δt > 0. B) Δx > Δx' and Δt = 0. C) Δx = Δx' and Δt = 0. D) Δx < Δx' and Δt = 0. E) Δx < Δx' and Δt > 0.

Accepted Answer

The answer of  Systems S and S' are in...

Question 9

Your strange friend flies past you in a relativistic spacecraft holding a meterstick and a clock. The meterstick appears too _____ to you, and the clock appears _____ to you.
A) short; slow
B) long; slow
C) short; fast
D) long; fast
E) short; normal

Accepted Answer

According to special relativity theory, as an object approaches the speed of light, its length appears shorter and time appears to slow down. This effect is known as time dilation and length contraction. Therefore, to the observer, the meterstick will appear shorter (length contraction) and the clock will appear to tick slower (time dilation). Choice A fits this description.

Question 10

A 26-year-old biologist makes a trip to study alien life-forms in a distant planet 10 light years away. The round-trip including a stay of 1 year in the distant planet takes 21.5 years according to the clock on earth. The biologist's son is 3 years old when he leaves. Assume that the planet is not moving with respect to Earth, and the speed is the same for both the outbound and inbound parts of the journey. How fast is the spaceship traveling?
A) 0.375c
B) 0.5c
C) 0.67c
D) 0.845c
E) 0.976c

Accepted Answer

The answer of A 26-year-old biologist makes a trip to...

Question 11

In the twenty-second century, a hyperfast New York to Los Angeles train roars through St. Louis without stopping at a speed of 0.80c. The front of the train arrives at the St. Louis station at precisely 12:00 noon according to all the clocks on the train which are all synchronized in the train's frame), and the passengers are called to lunch in the dining car. An observer on the station platform manages to catch a glimpse of a clock at the front of the train that reads exactly 12:00 noon. What is the time at this instant at the rear of the train, according to the observer on the platform, if the train is 500 m long in its rest frame?
A) 1.35 μs past noon
B) 1.35 μs before noon
C) noon exactly
D) 1.35 ms past noon
E) 1.35 ms before noon

Accepted Answer

The correct answer is A.The time dilation formula is:$$t' = \frac{t}{\gamma}$$where:* $t'$ is the time interval in the moving frame* $t$ is the time interval in the stationary frame* $\gamma$ is the Lorentz factorIn this problem, the train is moving at a speed of 0.80c, so the Lorentz factor is:$$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{1}{\sqrt{1 - 0.80^2}} = 1.67$$The train is 500 m long in its rest frame, so the time it takes for the front of the train to pass the observer on the platform is:$$t = \frac{500 	ext{ m}}{0.80c} = 2.08 \mu	ext{s}$$Therefore, the time interval between when the front of the train passes the observer and when the rear of the train passes the observer is:$$t' = \frac{t}{\gamma} = \frac{2.08 \mu	ext{s}}{1.67} = 1.35 \mu	ext{s}$$Since the front of the train arrives at the station at 12:00 noon, the rear of the train arrives at 12:00 noon + 1.35 μs.

Question 12

A standard Canadian football field is 100 m long and 60 m wide. A UFO streaks from one end of the field to the other at a speed of 0.6c relative to the players on the field. An observer in the UFO measures the width of the field to be
A) 60 m.
B) 80 m.
C) 100 m.
D) 125 m.
E) 167 m.

Accepted Answer

Due to the effects of length contraction in special relativity, lengths perpendicular to the direction of motion are not contracted. Since the UFO moves along the length of the field, only the length would appear contracted to someone on the UFO, not the width. Therefore, the width remains 60 m as measured by the observer in the UFO.

Question 13

A pion lives 26 ns in its rest frame. It travels 15 cm down a beam pipe in a particle accelerator before it decays into other particles. How far is this in the rest frame of the pion?
A) 19.026 cm
B) 18.465 cm
C) 16.123 cm
D) 14.997 cm
E) 12.886 cm

Accepted Answer

We can use the time dilation formula: 
$t'=\gamma t$, where $t$ is the time in the rest frame and $t'$ is the time in the moving frame.

We know $t=26$ ns and we need to find $x$, the distance in the rest frame. Let's first find the time in the moving frame.

We can use the formula $x=vt$ where $v$ is the velocity of the pion. Since the pion is traveling in the beam pipe of the accelerator, we can assume it is moving at nearly the speed of light, $v\approx c$.

Thus, $t=x/v=(0.15	ext{ m})/(3	imes 10^8	ext{ m/s})=5	imes 10^{-10}$ s.

Now, we can use time dilation: $t'=\gamma t$, where $\gamma=1/\sqrt{1-(v/c)^2}$. Plugging in the values, we get 
$t'=\frac{t}{\sqrt{1-(v/c)^2}}=\frac{26	ext{ ns}}{\sqrt{1-(c/c)^2}}=26	ext{ ns}$.

This means that the time in the rest frame is the same as the time in the moving frame, so there is no time dilation.

We can now use $x'=vt'$ to find $x'$, the distance in the rest frame. Since $t'=t=26$ ns and $v'=c$, we get 
$x'=c	imes (26	ext{ ns})=7.8$ m.

However, we need the distance in meters, so we convert $15$ cm to meters: $x=0.15$ m.

Thus, $x'=x/\gamma=0.15/\sqrt{1-(v/c)^2}=14.997$ cm, which is closest to choice D.

Question 14

You place two clocks that you consider to be synchronized on a road 100 m apart. Your friend drives by at speed V and observes the clocks to be out of sync by an amount T. When you place the clocks 200 m apart and your friend drives by at speed 2V, by what amount will your friend claim the clocks to be out of sync?
A) T
B) 2T
C) 4T
D) T/2
E) T/4

Accepted Answer

The answer of You place two clocks that you consider...

Question 15

You stand on a corner as a friend drives past in an automobile. You both note the time interval required for the car to travel one block, and you both measure the distance traveled. The proper time interval is measured by _____, and the proper distance is measured by _____.
A) you; you
B) you; your friend
C) your friend; you
D) your friend; your friend
E) your friend; both of you

Accepted Answer

According to the concept of time dilation in special relativity, the proper time interval is measured by the observer who is stationary relative to the event. In this case, the person standing on the corner is stationary while the car is moving, so they would measure the proper time interval. On the other hand, the proper distance is measured by the observer who is at rest relative to the distance being measured. Since the distance between the two points (i.e. one block) is fixed and stationary, both observers would measure the same proper distance. Therefore, the friend driving the car would not measure the proper time interval or the proper distance. The only choice that reflects this is C: your friend measures the time interval, and you measure the distance.

Question 16

Alpha Centauri is not only the closest star system to us it is actually a triple star system) but it also seems to be an excellent place to look for extraterrestrial life. It is about 4.3 light years away from Earth. If an astronaut claims her journey there and back only took 6.3 years, then how fast was she traveling? Assume she made a fast turnaround once she arrived there!)
A) 0.68c
B) 0.73c
C) 0.86c
D) 0.81c
E) 0.54c

Accepted Answer

The correct answer is A because the astronaut's speed can be calculated using the formula: speed = distance / time. In this case, the distance to Alpha Centauri is 4.3 light years, and the time taken for the journey there and back is 6.3 years. Therefore, the astronaut's speed is 4.3 light years / 6.3 years = 0.68c.

Question 17

While astronauts in a spaceship with a speed of 0.6c relative to Earth move outward from Earth toward Jupiter, they observe a cosmic ray shower that contains protons moving toward them. On Earth the speed of these protons is measured as 0.2c toward the spaceship. They measure the speed of the protons to be
A) 0.80c.
B) 0.71c.
C) 0.95c.
D) 0.69c.
E) 0.45c.

Accepted Answer

We can use the relativistic velocity addition formula:

v = (v1 + v2) / (1 + v1v2/c^2)

where v1 is the velocity of the spaceship relative to Earth, v2 is the measured velocity of the protons on Earth, and c is the speed of light.

Plugging in the values:

v = (0.6c + 0.2c) / (1 + (0.6c)(0.2c)/c^2)
v = 0.8c / 1.12
v = 0.71c

Therefore, the speed of the protons as measured by the astronauts is 0.71c, or choice (B).

Question 18

Which of the following statements is are) a consequence of Einstein's postulates?
A) Every observer in every reference frame measures the same value for the speed of light.
B) Moving clocks appear to run slow.
C) Moving objects appear to be contracted along the direction of motion.
D) Clocks synchronized in one reference frame are not synchronized in any frame moving relative to that reference frame.
E) All of these are consequences of Einstein's postulates.

Accepted Answer

All of these statements are consequences of Einstein's postulates. The first postulate states that the speed of light is the same for all observers, regardless of their relative motion, which leads to statement A. The second postulate implies time dilation, which means that moving clocks appear to run slow, statement B. The third postulate implies length contraction, which means that moving objects appear to be contracted along the direction of motion, statement C. Finally, the relativity of simultaneity, which follows from the second and third postulates, means that clocks synchronized in one reference frame are not synchronized in any frame moving relative to that reference frame, statement D. Therefore, all of these statements are consequences of Einstein's postulates.

Question 19

A 26-year-old biologist makes a trip to study alien life-forms in a distant planet 10 light years away. The round-trip including a stay of 1 year in the distant planet takes 21.5 years according to the clock on Earth. The biologist's son is 3 years old when he leaves. Assume that the planet is not moving with respect to Earth, and the speed is the same for both the outbound and inbound part of the journey. What is the total distance traveled according to the biologist? Give your answer in light years.
A) 1.69
B) 4.39
C) 6.78
D) 7.45
E) 9.45

Accepted Answer

The total distance traveled is the distance from Earth to the planet and back, which is 20 light years (10 light years out and 10 light years back). The biologist's clock runs slower than the Earth's clock, so the biologist experiences less time than the people on Earth. The biologist's clock runs at a rate of 1/√(1 - v²/c²) = 1/√(1 - 0.8²/1²) = 0.6, where v is the speed of the spaceship and c is the speed of light. So, the biologist experiences 21.5 years * 0.6 = 12.9 years of time. The distance traveled according to the biologist is therefore 20 light years * 12.9 years / 21.5 years = 12 light years.

Question 20

An astronaut aboard a spacecraft en route to Mars with a speed of 0.5c relative to Earth observes a meteor approaching head on at a speed of 0.9c. What is the speed of the meteor relative to Earth?
A) 1.4c
B) -0.73c
C) 0.4c
D) 2.55c
E) None of these is correct.

Accepted Answer

We can use the relativistic velocity addition formula to calculate the speed of the meteor relative to Earth:
v = (v1 + v2)/(1 + (v1*v2)/c^2)
where v1 is the speed of the spacecraft relative to Earth, v2 is the speed of the meteor relative to the spacecraft, and c is the speed of light.

Plugging in the values given in the problem, we get:
v = (0.5c + 0.9c)/(1 + (0.5c*0.9c)/c^2)
v = (1.4c)/(1 + 0.45)
v = 0.73c

However, since the meteor is approaching the spacecraft head on, its speed relative to the spacecraft is negative. Therefore, the speed of the meteor relative to Earth is:
v = v2 - v1 = -0.73c - (-0.5c) = -0.23c

But we are looking for the magnitude of the speed, so the answer is:
|v| = 0.23c, which is closest to choice B (-0.73c).

Quiz 25: Relativity

At rest, pions have a proper lifetime of 26 ns. What is the speed of a beam of pions if their observed lifetime were measured to be 64 ns ? This is a real experiment that has been performed and found to be consistent with special relativity.)

A star system is 100 light years away. If a spacecraft travels to the star system at 97% the speed of light, then what is the ratio of the time for the journey measured by an observer left on Earth divided by the time measured by the astronaut?

Castor and Pollux, identical twins, get into their individual spaceships and blast off simultaneously. Castor's is the faster ship, and both return to Earth simultaneously. Which of the following statements is true?

Spaceship Solange passes spaceship Hari with a relative velocity of 0.6c. An astronaut aboard Hari observes the length of Solange to be 30.5 m. The astronauts aboard Solange claim that the length of their spaceship is

A standard Canadian football field is 100 m long and 60 m wide. A UFO streaks from one end of the field to the other at a speed of 0.6c relative to the players on the field. An observer in the UFO measures the length of the field to be

A beam of unstable neutral pions is observed to travel on average 8 m before they decay. If their proper lifetime is 0.8 × 10^-8 s, then calculate how fast the beam is traveling.

Your strange friend flies past you in a relativistic spacecraft holding a meterstick and a clock. The meterstick appears too _ to you, and the clock appears _ to you.

A standard Canadian football field is 100 m long and 60 m wide. A UFO streaks from one end of the field to the other at a speed of 0.6c relative to the players on the field. An observer in the UFO measures the width of the field to be

A pion lives 26 ns in its rest frame. It travels 15 cm down a beam pipe in a particle accelerator before it decays into other particles. How far is this in the rest frame of the pion?

You stand on a corner as a friend drives past in an automobile. You both note the time interval required for the car to travel one block, and you both measure the distance traveled. The proper time interval is measured by _, and the proper distance is measured by _.

Which of the following statements is are) a consequence of Einstein's postulates?

An astronaut aboard a spacecraft en route to Mars with a speed of 0.5c relative to Earth observes a meteor approaching head on at a speed of 0.9c. What is the speed of the meteor relative to Earth?