Question 1

A 101-meter long relativistic train approaches a 100-m long platform at high speed.How fast must the train be going to fit entirely inside the length of the platform?

Accepted Answer

To fit a 101-meter long train entirely inside a 100-meter platform due to relativistic length contraction, the length contraction formula is used: $L = L_0\sqrt{1 - \frac{v^2}{c^2}}$, where $L$ is the contracted length (100 meters), $L_0$ is the rest length of the train (101 meters), and $v$ is the velocity of the train. Solving for $v$, we find that the speed must be significant enough to cause a noticeable contraction, which corresponds to choice B when calculated correctly.

Question 2

Right before takeoff,a passenger on a plane flying from town A to town B synchronizes his clock with the clock of his friend waiting for him in town B. The plane flies without interruption for 5.00 hours at a constant velocity.The moment the plane touches the ground the two friends check simultaneously the indication of their clocks.The difference between the readings of the clocks B and A is 7.56 ns.Ignoring any effects of acceleration,the constant velocity of the plane was

Accepted Answer

E

Question 3

A spacecraft travels along a straight line from the Earth to the Moon a distance of 3.84 * 10⁸ m.Its speed measured on Earth is 0.70 c.How long does the trip take according to a clock on the spacecraft?

Accepted Answer

The time taken according to a clock on the spacecraft can be calculated using the time dilation formula in special relativity, $ t' = \frac{t}{\gamma} $, where $ t $ is the time in the stationary frame (Earth), $ \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} $ is the Lorentz factor, and $ v $ is the velocity of the spacecraft. First, calculate the time in the Earth frame: $ t = \frac{distance}{speed} = \frac{3.84 \times 10^{8}}{0.70c} $. Since $ c = 3.00 \times 10^{8} \, m/s $, $ t = \frac{3.84 \times 10^{8}}{0.70 \times 3.00 \times 10^{8}} $ seconds. Then, calculate $ \gamma $ and find $ t' $. The correct calculation will show that the time dilation effect makes the trip appear shorter on the spacecraft, leading to the answer of approximately 1.3 seconds.

Question 4

An observer on the Earth sees the U.S.S.Enterprise approaching from one direction at 0.600 c and a Klingon Battle Cruiser approaching from the opposite direction at 0.800 c.According to Mr.Spock,who is aboard the Enterprise,what is the speed of the Klingon Battle Cruiser?

Accepted Answer

According to the relativistic velocity addition formula,v_total=(v1+v2)/(1+(v1*v2)/c^2), the relative velocity of the Klingon Battle Cruiser with respect to the observer on Earth is v= (0.800c+(-0.600c))/(1+(0.800c*(-0.600c))/c^2)= 1.40 c/(1-0.48)=0.95 c. Therefore, the correct choice is C.

Question 5

A spacecraft travels along a straight line from the Earth to the Moon a distance of 3.84 * 10⁸ m.Its speed measured on Earth is 0.70 c.How long does the trip take according to a clock on Earth?

Accepted Answer

To find the time taken according to a clock on Earth, we use the formula $t = \frac{d}{v}$, where $d = 3.84 \times 10^{8} \, \text{m}$ is the distance from the Earth to the Moon, and $v = 0.70c$ is the speed of the spacecraft with $c = 3.00 \times 10^{8} \, \text{m/s}$ being the speed of light. Substituting the values, we get $t = \frac{3.84 \times 10^{8} \, \text{m}}{0.70 \times 3.00 \times 10^{8} \, \text{m/s}} = 1.83 \, \text{s}$, which rounds to 1.8 seconds.

Question 6

Two observers standing upright on their feet,move horizontally past each other and look at each other.The acceleration of both is zero and their relative speed is 0.500 c.Observer one looks at observer two and sees that she looks thinner than she is when she isn't moving.What does observer two see?

Accepted Answer

This phenomenon is known as the relativistic length contraction, where the length of an object appears shorter when it is moving relative to the observer. As observer one is moving relative to observer two, she appears thinner to observer one. Similarly, observer two would also see observer one as thinner than she is when she isn't moving.

Question 7

An astronaut on the space shuttle traveling at a speed of 0.70 c is holding a meter stick parallel to the direction of motion.If an observer on Earth could observe the meter stick,what length of the meter stick would he measure?

Accepted Answer

The length of the meter stick would appear contracted to an observer on Earth due to the phenomenon of length contraction in special relativity. The formula for length contraction is L = L0 * sqrt(1 - v^2/c^2), where L0 is the rest length (1 meter in this case), v is the velocity of the object (0.70c), and c is the speed of light. Plugging in the values, we find that the observed length L is less than the rest length, which matches closest to option A, 0.71 m.

Question 8

How much energy is released when a neutron (rest mass 939.6 MeV/c²)decays into a proton (rest mass 938.3 MeV/c²),an electron (rest mass 0.511 MeV/c²)and an anti-electron neutrino (rest mass approximately zero)?

Accepted Answer

The energy released is the difference in rest mass energy between the initial and final particles: 939.6 MeV - (938.3 MeV + 0.511 MeV) = 0.789 MeV or 789 keV.

Question 9

An observer on the Earth sees the U.S.S.Enterprise approaching from one direction at 0.60 c and a Klingon Battle Cruiser approaching from the opposite direction.According to Mr.Spock,who is aboard the Enterprise,the Klingon Battle Cruiser is approaching the Enterprise,that 0.80 c.What is the speed of the Klingon Battle Cruiser as observed from Earth?

Accepted Answer

We can use the relativistic velocity addition formula to solve this problem:
v = (v1 + v2) / (1 + v1v2/c^2)

Where:

v1 = 0.60 c (velocity of the Enterprise as observed from Earth)
v2 = -0.80 c (velocity of the Klingon Battle Cruiser as observed from the Enterprise)
c = 3.00 x 10^8 m/s (speed of light)

Plugging in the values, we get:

v = (0.60 c - 0.80 c) / (1 - (0.60 c)(-0.80 c)/c^2) = 0.38 c

Therefore, the speed of the Klingon Battle Cruiser as observed from Earth is 0.38 c, which corresponds to choice D).

Question 10

Sam sees two events as simultaneous: (i)event A occurs at the point (0,0,0)at the instant 0:00:00 Universal time;
(ii.event B occurs at the point (569 m,0,0)at the same moment.
Tim,moving past Sam with a velocity of (0.998 c)",also observes the two events.Relative to the time Tim measures for event A,when does Tim see event B occur?

Accepted Answer

The answer of Sam sees two events as simultaneous: (i)event...

Question 11

A new particle is discovered at Stanford Linear Accelerator.If the total energy of the particle is 17.0 GeV and its momentum is 5.00 GeV/c,what is the rest mass of the particle?

Accepted Answer

We can use the formula for relativistic momentum:

p = γmv

where γ is the Lorentz factor, m is the rest mass, and v is the velocity.

We know that the total energy (E) of the particle is

E = γmc^2 + K

where c is the speed of light and K is the kinetic energy. Since the particle's momentum is non-zero, we know that K is also non-zero, so

E > γmc^2

Solving for γ:

γ = E/(mc^2) > 1

We're given the values of E and p, so we can solve for γ:

γ = E/√(p^2c^2 + m^2c^4)

Squaring both sides:

γ^2 = E^2/(p^2c^2 + m^2c^4)

Rearranging:

m^2c^4 = E^2/γ^2 - p^2c^2

Plugging in the values:

m^2c^4 = (17.0 GeV)^2/(γ^2) - (5.00 GeV/c)^2

Since γ > 1, we know that m < E/c^2. The only answer choice that satisfies that condition is B.

Question 12

A spacecraft travels along a straight line from the Earth to the Moon a distance of 3.84 * 10⁸ m.Its speed measured on Earth is 0.70 c.Determine the distance between the Earth and Moon if it were measured by a person on the spacecraft.

Accepted Answer

Let's use the formula for length contraction:

L' = L / γ

where L is the length as measured on Earth, L' is the length as measured on the spacecraft, and γ is the Lorentz factor given by:

γ = 1 / sqrt(1 - v^2/c^2)

where v is the speed of the spacecraft and c is the speed of light.

Plugging in the numbers, we get:

γ = 1 / sqrt(1 - 0.70^2) = 1.71

L' = 3.84 x 10^8 m / 1.71 = 2.24 x 10^8 m

So the distance between the Earth and Moon as measured by a person on the spacecraft is 2.24 x 10^8 m, or 2.7 x 10^8 m (rounded to one significant figure). Therefore, the best choice is C.

Question 13

A new particle is discovered at Stanford Linear Accelerator whose rest mass is 12.0 GeV/c².If the total energy of the particle is 17.0 GeV,what is the momentum of this new particle?

Accepted Answer

The answer of A new particle is discovered at Stanford...

Question 14

Two observers move past each other and look at each other's clock.The acceleration of both is zero and their relative speed is 0.750 c.Observer one looks at observer two's clock and sees that it's running more slowly than hers.What does observer two see?

Accepted Answer

According to Einstein's theory of relativity, time dilation occurs when two observers are moving at different speeds relative to each other. In this scenario, observer one is moving relative to observer two, and therefore experiences time dilation. This means that observer two sees observer one's clock running more slowly than his own clock. Therefore, observer two would choose option B.

Question 15

Sam sees two events as simultaneous: i.event A occurs at the point (0,0,0)at the instant 0:00:00 Universal time; ii.event B occurs at the point (x_B,0,0)at the same moment. Tim,moving past Sam with a velocity of (0.998 c)",also observes the two events and sees that event B happened 3.74 µs after event A. Then x_B is

Accepted Answer

The correct answer is A because the time dilation formula is:Δt' = Δt / γwhere Δt' is the time interval measured by the moving observer, Δt is the time interval measured by the stationary observer, and γ is the Lorentz factor. In this case, Δt' = 3.74 µs and Δt = 0 s, so γ = 1 / Δt' = 1 / 3.74 µs = 267.4. The Lorentz factor is given by:γ = 1 / sqrt(1 - v^2 / c^2)where v is the velocity of the moving observer and c is the speed of light. In this case, v = 0.998 c, so:γ = 1 / sqrt(1 - 0.998^2) = 267.4The distance between the two events is given by:x = vΔtwhere x is the distance between the two events, v is the velocity of the moving observer, and Δt is the time interval measured by the moving observer. In this case, v = 0.998 c and Δt = 3.74 µs, so:x = 0.998 c * 3.74 µs = 71.1 mTherefore, the correct answer is A.

Question 16

Right before takeoff,a passenger on a plane flying from town A to town B synchronizes his clock with the clock of his friend waiting for him in town B. The plane flies without interruption at a constant velocity of 280 m/s. The moment the plane touches the ground the two friends check simultaneously the indication of their clocks.The clock of the passenger on the plane shows that it takes exactly 7.50 h to travel from A to B. Ignoring any effects of acceleration,the difference between the readings of the clocks B and A is

Accepted Answer

The time dilation effect due to the plane's velocity causes the clock on the plane to run slower than the clock on the ground. The difference in time, calculated using the formula for time dilation in special relativity, is approximately 11.8 nanoseconds.

Question 17

Right before takeoff,a passenger on a plane flying from town A to town B synchronizes his clock with the clock of his friend waiting for him in town B. The plane flies without interruption at a constant velocity of 285 m/s.The moment the plane touches the ground the two friends check simultaneously the indication of their clocks.The difference between the readings of the clocks B and A is 4.88 ns.Ignoring any effects of acceleration,the clock A of the passenger on the plane indicated that the flight took

Accepted Answer

The answer of Right before takeoff,a passenger on a plane...

Question 18

A new particle is discovered at Stanford Linear Accelerator whose rest mass is 12.0 GeV/c².If the momentum of this new particle was 5.00 GeV/c,then what would be its total energy?

Accepted Answer

The total energy of a particle with rest mass m and momentum p is given by the relativistic energy equation:

E = √(p^2c^2 + m^2c^4)

where c is the speed of light.

Substituting the given values, we get:

E = √((5.00 GeV/c)^2c^2 + (12.0 GeV/c^2)^2c^4)

= √(25.0 GeV^2 + 144.0 GeV^2)

= √169.0 GeV^2

= 13.0 GeV

Therefore, the total energy of the particle is 13.0 GeV. The correct choice is C.

Question 19

An observer on the Earth sees the U.S.S.Enterprise approaching from one direction at 0.600 c and a Klingon Battle Cruiser approaching from the opposite direction at 0.800 c.According to Mr.Spock,who is aboard the Enterprise,what is the speed of the Klingon Battle Cruiser?

Accepted Answer

The correct answer involves using the formula for relativistic velocity addition. According to this formula, the velocity of the Klingon Battle Cruiser relative to the Enterprise, as observed by Mr. Spock, is not simply the sum of the two speeds (which would suggest an impossible speed greater than c). Instead, the formula must be applied, which ensures that the resulting speed is less than the speed of light (c). The options provided do not match the result of this calculation, hence none of the given choices are correct.

Question 20

An astronaut on the space shuttle traveling at a speed of 0.70 c is holding a meter stick parallel to the direction of motion.What is the length of the meter stick as measured by another astronaut on the shuttle?

Accepted Answer

The length of the meter stick as measured by another astronaut on the shuttle would remain 1.0 m because both the observer and the object (meter stick) are in the same inertial frame of reference, meaning there is no relative motion between them that would cause length contraction.

Quiz 35: Relativity

A 101-meter long relativistic train approaches a 100-m long platform at high speed.How fast must the train be going to fit entirely inside the length of the platform?

A spacecraft travels along a straight line from the Earth to the Moon a distance of 3.84 * 10⁸ m.Its speed measured on Earth is 0.70 c.How long does the trip take according to a clock on the spacecraft?

An observer on the Earth sees the U.S.S.Enterprise approaching from one direction at 0.600 c and a Klingon Battle Cruiser approaching from the opposite direction at 0.800 c.According to Mr.Spock,who is aboard the Enterprise,what is the speed of the Klingon Battle Cruiser?

A spacecraft travels along a straight line from the Earth to the Moon a distance of 3.84 * 10⁸ m.Its speed measured on Earth is 0.70 c.How long does the trip take according to a clock on Earth?

Two observers standing upright on their feet,move horizontally past each other and look at each other.The acceleration of both is zero and their relative speed is 0.500 c.Observer one looks at observer two and sees that she looks thinner than she is when she isn't moving.What does observer two see?

An astronaut on the space shuttle traveling at a speed of 0.70 c is holding a meter stick parallel to the direction of motion.If an observer on Earth could observe the meter stick,what length of the meter stick would he measure?

How much energy is released when a neutron (rest mass 939.6 MeV/c²) decays into a proton (rest mass 938.3 MeV/c²) ,an electron (rest mass 0.511 MeV/c²) and an anti-electron neutrino (rest mass approximately zero) ?

A new particle is discovered at Stanford Linear Accelerator.If the total energy of the particle is 17.0 GeV and its momentum is 5.00 GeV/c,what is the rest mass of the particle?

A spacecraft travels along a straight line from the Earth to the Moon a distance of 3.84 * 10⁸ m.Its speed measured on Earth is 0.70 c.Determine the distance between the Earth and Moon if it were measured by a person on the spacecraft.

A new particle is discovered at Stanford Linear Accelerator whose rest mass is 12.0 GeV/c².If the total energy of the particle is 17.0 GeV,what is the momentum of this new particle?

Two observers move past each other and look at each other's clock.The acceleration of both is zero and their relative speed is 0.750 c.Observer one looks at observer two's clock and sees that it's running more slowly than hers.What does observer two see?

A new particle is discovered at Stanford Linear Accelerator whose rest mass is 12.0 GeV/c².If the momentum of this new particle was 5.00 GeV/c,then what would be its total energy?

An observer on the Earth sees the U.S.S.Enterprise approaching from one direction at 0.600 c and a Klingon Battle Cruiser approaching from the opposite direction at 0.800 c.According to Mr.Spock,who is aboard the Enterprise,what is the speed of the Klingon Battle Cruiser?

An astronaut on the space shuttle traveling at a speed of 0.70 c is holding a meter stick parallel to the direction of motion.What is the length of the meter stick as measured by another astronaut on the shuttle?

Quiz 35: Relativity

A 101-meter long relativistic train approaches a 100-m long platform at high speed.How fast must the train be going to fit entirely inside the length of the platform?

A spacecraft travels along a straight line from the Earth to the Moon a distance of 3.84 * 108 m.Its speed measured on Earth is 0.70 c.How long does the trip take according to a clock on the spacecraft?

An observer on the Earth sees the U.S.S.Enterprise approaching from one direction at 0.600 c and a Klingon Battle Cruiser approaching from the opposite direction at 0.800 c.According to Mr.Spock,who is aboard the Enterprise,what is the speed of the Klingon Battle Cruiser?

A spacecraft travels along a straight line from the Earth to the Moon a distance of 3.84 * 108 m.Its speed measured on Earth is 0.70 c.How long does the trip take according to a clock on Earth?

Two observers standing upright on their feet,move horizontally past each other and look at each other.The acceleration of both is zero and their relative speed is 0.500 c.Observer one looks at observer two and sees that she looks thinner than she is when she isn't moving.What does observer two see?

An astronaut on the space shuttle traveling at a speed of 0.70 c is holding a meter stick parallel to the direction of motion.If an observer on Earth could observe the meter stick,what length of the meter stick would he measure?

How much energy is released when a neutron (rest mass 939.6 MeV/c2) decays into a proton (rest mass 938.3 MeV/c2) ,an electron (rest mass 0.511 MeV/c2) and an anti-electron neutrino (rest mass approximately zero) ?

A new particle is discovered at Stanford Linear Accelerator.If the total energy of the particle is 17.0 GeV and its momentum is 5.00 GeV/c,what is the rest mass of the particle?

A spacecraft travels along a straight line from the Earth to the Moon a distance of 3.84 * 108 m.Its speed measured on Earth is 0.70 c.Determine the distance between the Earth and Moon if it were measured by a person on the spacecraft.

A new particle is discovered at Stanford Linear Accelerator whose rest mass is 12.0 GeV/c2.If the total energy of the particle is 17.0 GeV,what is the momentum of this new particle?

Two observers move past each other and look at each other's clock.The acceleration of both is zero and their relative speed is 0.750 c.Observer one looks at observer two's clock and sees that it's running more slowly than hers.What does observer two see?

A new particle is discovered at Stanford Linear Accelerator whose rest mass is 12.0 GeV/c2.If the momentum of this new particle was 5.00 GeV/c,then what would be its total energy?

An observer on the Earth sees the U.S.S.Enterprise approaching from one direction at 0.600 c and a Klingon Battle Cruiser approaching from the opposite direction at 0.800 c.According to Mr.Spock,who is aboard the Enterprise,what is the speed of the Klingon Battle Cruiser?

An astronaut on the space shuttle traveling at a speed of 0.70 c is holding a meter stick parallel to the direction of motion.What is the length of the meter stick as measured by another astronaut on the shuttle?

A spacecraft travels along a straight line from the Earth to the Moon a distance of 3.84 * 10⁸ m.Its speed measured on Earth is 0.70 c.How long does the trip take according to a clock on the spacecraft?

A spacecraft travels along a straight line from the Earth to the Moon a distance of 3.84 * 10⁸ m.Its speed measured on Earth is 0.70 c.How long does the trip take according to a clock on Earth?

How much energy is released when a neutron (rest mass 939.6 MeV/c²) decays into a proton (rest mass 938.3 MeV/c²) ,an electron (rest mass 0.511 MeV/c²) and an anti-electron neutrino (rest mass approximately zero) ?

A spacecraft travels along a straight line from the Earth to the Moon a distance of 3.84 * 10⁸ m.Its speed measured on Earth is 0.70 c.Determine the distance between the Earth and Moon if it were measured by a person on the spacecraft.

A new particle is discovered at Stanford Linear Accelerator whose rest mass is 12.0 GeV/c².If the total energy of the particle is 17.0 GeV,what is the momentum of this new particle?

A new particle is discovered at Stanford Linear Accelerator whose rest mass is 12.0 GeV/c².If the momentum of this new particle was 5.00 GeV/c,then what would be its total energy?