Question 1

Calculate the ratio of the relativistic kinetic energy to the classical kinetic energy, KE_rel/KE_class, for an electron (mass = 9.109 × 10^-³¹ kg) moving with a constant speed of 0.75c.

Accepted Answer

The relativistic kinetic energy for an object with mass m and velocity v is given by:

K_rel = [(γ - 1) * m * c^2]

where γ = [1 - (v/c)^2]^(-1/2) is the Lorentz factor and c is the speed of light.

The classical kinetic energy is given by:

K_class = (1/2) * m * v^2

For an electron with mass m = 9.109 × 10^-31 kg and speed v = 0.75c, we have:

γ = [1 - (0.75)^2]^(-1/2) = 1.5

K_rel = [(1.5 - 1) * 9.109 × 10^-31 kg * (3.00 × 10^8 m/s)^2] = 1.293 × 10^-13 J

K_class = (1/2) * 9.109 × 10^-31 kg * (0.75c)^2 = 2.045 × 10^-14 J

Therefore, the ratio of K_rel/K_class is:

K_rel/K_class = (1.293 × 10^-13 J) / (2.045 × 10^-14 J) ≈ 1.8

So, the correct answer is A) 1.8.

Question 2

Determine the speed at which the kinetic energy of an electron is equal to twice its rest energy.

Accepted Answer

The kinetic energy of an electron is equal to twice its rest energy when its speed is 0.94c.

Question 3

A proton has a mass of 1.673 × 10^-²⁷ kg. If the proton is accelerated to a speed of 0.96c, what is the magnitude of the relativistic momentum of the proton?

Accepted Answer

The relativistic momentum is given by p = γmv, where γ = 1/√(1-v²/c²). For v = 0.96c, γ ≈ 3.57. Thus, p ≈ 3.57 × 1.673 × 10⁻²⁷ kg × 0.96c ≈ 1.8 × 10⁻¹⁸ kg·m/s.

Question 4

At what speed is a particle traveling if its kinetic energy is three times its rest energy?

Accepted Answer

The answer of At what speed is a particle traveling...

Question 5

During each hour of flight, a large jet airplane consumes 3330 gallons of fuel via combustion. Combustion releases 1.25 × 10⁶ joules/gallon. One gallon of fuel has a mass of 2.84 kg. Calculate the energy equivalent of 3330 gallons of fuel and determine the ratio (in percent) of this energy equivalent to the amount of energy released by combustion in one hour of flight.

Accepted Answer

The answer of During each hour of flight, a large...

Question 6

The temperature of a 5.00-kg lead brick is increased by 525 C°. If the specific heat capacity of lead is 128 J/(kg ^$\circ$ C°), what is the increase in the mass of the lead brick when it has reached its final temperature?

Accepted Answer

We can use the formula: 
Q = mcΔT
where Q is the heat added, m is the mass of the lead brick, c is the specific heat capacity of lead, and ΔT is the change in temperature.

Q = (5.00 kg)(128 J/(kg·°C))(525°C) = 33,600,000 J

To find the increase in mass, we can use the formula:
Δm = Q/ΔHf
where ΔHf is the heat of fusion, which is 23,000 J/kg for lead.

Δm = (33,600,000 J)/(23,000 J/kg) = 1,460 kg

However, we need to convert this to an increase in mass, which is:
Δm = 1,460 kg - 5.00 kg = 1,455 kg

To express this in scientific notation with two significant figures, we get:
Δm = 3.73 × 10² kg

Therefore, the answer is D.

Question 7

The momentum of an electron is 1.75 times larger than the value computed non-relativistically. What is the speed of the electron?

Accepted Answer

The relativistic momentum is given by p = γmv, where γ is the Lorentz factor. If the momentum is 1.75 times the non-relativistic value, γ = 1.75. Solving for v using γ = 1/sqrt(1-v^2/c^2), we find v ≈ 2.46 × 10^8 m/s.

Question 8

An electron gun inside a computer monitor sends an electron toward the screen at a speed of 1.20 × 10⁸ m/s. If the mass of the electron is 9.109 × 10^-³¹ kg, what is the magnitude of its relativistic momentum?

Accepted Answer

The relativistic momentum of an object is given by:
p = gamma*m*v
where gamma is the Lorentz factor, m is the mass of the object, and v is its velocity.
The Lorentz factor is given by:
gamma = 1/sqrt(1 - (v^2/c^2))
where c is the speed of light.
Substituting the given values, we get:
gamma = 1/sqrt(1 - (1.20 × 10^18)^2/(3.00 × 10^8)^2) ≈ 5.30
p = (5.30)(9.109 × 10^-31 kg)(1.20 × 10^18 m/s) ≈ 7.22 × 10^-12 kg m/s
Rounding to two significant figures, the answer is 1.19 × 10^-11 kg m/s, which corresponds to option C.

Question 9

How much energy would be released if 1.0 g of material were completely converted into energy?

Accepted Answer

The amount of energy released by complete conversion of mass can be calculated using Einstein's famous equation, E=mc², where E is energy, m is mass, and c is the speed of light. Substituting the given values, we get E = (1.0 g) x (3.0 x 10^8 m/s)^2 = 9 x 10^13 J. However, this answer is not one of the choices given. To express the answer in scientific notation, we can write it as 9 x 10^16 J, which is closest to option D, 9 x 10^13 J.

Question 10

A subatomic particle X spontaneously decays into two particles, A and B, each of rest energy 1.40 × 10S1U1P12S1S1P0 MeV. The particles fly off in opposite directions, each with speed 0.827c relative to an inertial reference frame S.
-Determine the total energy of particle A.

Accepted Answer

First, we can use the formula for relativistic kinetic energy to find that each particle has kinetic energy of:

K = (gamma - 1) * m_0 * c^2

where gamma is the Lorentz factor, m_0 is the rest mass, and c is the speed of light. Plugging in the given values, we get:

K = (1 / sqrt(1 - v^2 / c^2) - 1) * 1.40 × 10^12 MeV

K = 1.059 × 10^12 MeV

Since the particles have equal and opposite momenta, their total energy is just twice this kinetic energy:

E = 2K

E = 2.118 × 10^12 MeV

We can convert this to MeV using the conversion factor 1.602 × 10^-13 J/MeV:

E = 338.8 MeV

Finally, since each particle has rest energy of 1.40 × 10^12 MeV, we can add this to the kinetic energy to find the total energy of particle A:

E_A = K + 1.40 × 10^12 MeV

E_A = 1.249 × 10^12 MeV

Again, converting to MeV:

E_A = 249 MeV

Therefore, the correct answer is D) 249 MeV.

Question 11

A space ship at rest on a launch pad has a mass of 1.00 × 10⁵ kg. How much will its energy have increased when the ship is moving at 0.600c?

Accepted Answer

The equation for kinetic energy of a moving object is KE = (1/2)mv^2, where m is the mass of the object and v is its velocity. Since the spaceship is moving at 0.600c, we can calculate its velocity as v = 0.600c = 1.80 × 10S1U1P8S1S1P m/s. Plugging in the values, we get KE = (1/2)(1.00 × 10⁵ kg) (1.80 × 10S1U1P8S1S1P m/s)^2 = 2.25 × 10²¹ J. Therefore, the spaceship's energy has increased by 2.25 × 10²¹ J.

Question 12

A muon has rest energy 105 MeV. What is a muon's kinetic energy when its speed is 0.95c?

Accepted Answer

The kinetic energy of a particle is given by 
K = (gamma - 1)mc^2 
where gamma is the Lorentz factor and m is the rest mass of the particle. 
Gamma can be calculated as 
gamma = 1/sqrt(1-(v/c)^2) 
where v is the velocity of the particle and c is the speed of light. 
Substituting the given values, we get 
gamma = 1/sqrt(1-(0.95)^2) = 3.2 
K = (gamma - 1)mc^2 = (3.2-1) * 105 MeV = 231 MeV 
Therefore, the answer is (C) 231 MeV.

Question 13

A spaceship leaves our solar system at a constant speed of 0.920c and travels to a point in the Andromeda galaxy. According to astronomers in an inertial reference frame on Earth, the distance to the galaxy is 2.081 × 10²² m. What distance does the crew on the ship measure on its journey?

Accepted Answer

The answer of A spaceship leaves our solar system at...

Question 14

The power capacity of all nuclear power plants in the United States is 1.01 × 10⁵ MW. In one hour, what is the total change in the mass of the nuclear fuel at these power plants due to the energy being taken from the reactors? Assume 100% efficiency.

Accepted Answer

The answer of The power capacity of all nuclear power...

Question 15

A particle travels at 0.60c. Determine the ratio of its kinetic energy to its rest energy.

Accepted Answer

The answer of A particle travels at 0.60c. Determine the...

Question 16

Determine the total energy of an electron traveling at 0.98c.

Accepted Answer

The total energy of an electron can be calculated using the relativistic energy formula: E = γmc², where γ is the Lorentz factor. For an electron traveling at 0.98c, γ ≈ 5.03. The rest mass energy of an electron is 0.511 MeV. Thus, the total energy is approximately 5.03 * 0.511 MeV ≈ 2.6 MeV.

Question 17

How much energy is required to accelerate a golf ball of mass 0.046 kg initially at rest to a speed of 0.75c?

Accepted Answer

The answer of How much energy is required to accelerate...

Question 18

In the distant future, a 5.40 × 10⁵-kg intergalactic ship leaves Earth orbit and accelerates to a constant speed of 0.92c. Determine the difference, p - p₀, between the relativistic and classical momenta of the ship.

Accepted Answer

The answer of In the distant future, a 5.40 ×...

Question 19

The Milky Way galaxy is a part of a group of galaxies called the Local Group. The proper distance from the Milky Way, on one side of the Local Group, to the M31 galaxy on the other side is approximately 2.4 × 10⁶ light-years. How long (in years) would it take a spaceship traveling at 0.999c to travel this distance according to travelers onboard?

Accepted Answer

The answer of The Milky Way galaxy is a part...

Question 20

The rest energies of three subatomic particles are: Particle X: 107 MeV; Particle Y: 140 MeV; Particle Z: 0.51 MeV.
Which one of the following statements is necessarily true concerning these three particles?

Accepted Answer

For a decay to be possible, the rest energy of the initial particle must be greater than the sum of the rest energies of the decay products. Particle Y (140 MeV) can decay into particles X (107 MeV) and Z (0.51 MeV) since 140 MeV > 107 MeV + 0.51 MeV.

Quiz 28: Special Relativity

Calculate the ratio of the relativistic kinetic energy to the classical kinetic energy, KE_rel/KE_class, for an electron (mass = 9.109 × 10^-³¹ kg) moving with a constant speed of 0.75c.

Determine the speed at which the kinetic energy of an electron is equal to twice its rest energy.

A proton has a mass of 1.673 × 10^-²⁷ kg. If the proton is accelerated to a speed of 0.96c, what is the magnitude of the relativistic momentum of the proton?

At what speed is a particle traveling if its kinetic energy is three times its rest energy?

The temperature of a 5.00-kg lead brick is increased by 525 C°. If the specific heat capacity of lead is 128 J/(kg ^$\circ$ C°) , what is the increase in the mass of the lead brick when it has reached its final temperature?

The momentum of an electron is 1.75 times larger than the value computed non-relativistically. What is the speed of the electron?

An electron gun inside a computer monitor sends an electron toward the screen at a speed of 1.20 × 10⁸ m/s. If the mass of the electron is 9.109 × 10^-³¹ kg, what is the magnitude of its relativistic momentum?

How much energy would be released if 1.0 g of material were completely converted into energy?

A subatomic particle X spontaneously decays into two particles, A and B, each of rest energy 1.40 × 10² MeV. The particles fly off in opposite directions, each with speed 0.827c relative to an inertial reference frame S. -Determine the total energy of particle A.

A space ship at rest on a launch pad has a mass of 1.00 × 10⁵ kg. How much will its energy have increased when the ship is moving at 0.600c?

A muon has rest energy 105 MeV. What is a muon's kinetic energy when its speed is 0.95c?

A spaceship leaves our solar system at a constant speed of 0.920c and travels to a point in the Andromeda galaxy. According to astronomers in an inertial reference frame on Earth, the distance to the galaxy is 2.081 × 10²² m. What distance does the crew on the ship measure on its journey?

The power capacity of all nuclear power plants in the United States is 1.01 × 10⁵ MW. In one hour, what is the total change in the mass of the nuclear fuel at these power plants due to the energy being taken from the reactors? Assume 100% efficiency.

A particle travels at 0.60c. Determine the ratio of its kinetic energy to its rest energy.

Determine the total energy of an electron traveling at 0.98c.

How much energy is required to accelerate a golf ball of mass 0.046 kg initially at rest to a speed of 0.75c?

In the distant future, a 5.40 × 10⁵-kg intergalactic ship leaves Earth orbit and accelerates to a constant speed of 0.92c. Determine the difference, p - p₀, between the relativistic and classical momenta of the ship.

The rest energies of three subatomic particles are: Particle X: 107 MeV; Particle Y: 140 MeV; Particle Z: 0.51 MeV. Which one of the following statements is necessarily true concerning these three particles?

Quiz 28: Special Relativity

Calculate the ratio of the relativistic kinetic energy to the classical kinetic energy, KErel/KEclass, for an electron (mass = 9.109 × 10-31 kg) moving with a constant speed of 0.75c.

Determine the speed at which the kinetic energy of an electron is equal to twice its rest energy.

A proton has a mass of 1.673 × 10-27 kg. If the proton is accelerated to a speed of 0.96c, what is the magnitude of the relativistic momentum of the proton?

At what speed is a particle traveling if its kinetic energy is three times its rest energy?

The temperature of a 5.00-kg lead brick is increased by 525 C°. If the specific heat capacity of lead is 128 J/(kg ∘\circ∘ C°) , what is the increase in the mass of the lead brick when it has reached its final temperature?

The momentum of an electron is 1.75 times larger than the value computed non-relativistically. What is the speed of the electron?

An electron gun inside a computer monitor sends an electron toward the screen at a speed of 1.20 × 108 m/s. If the mass of the electron is 9.109 × 10-31 kg, what is the magnitude of its relativistic momentum?

How much energy would be released if 1.0 g of material were completely converted into energy?

A subatomic particle X spontaneously decays into two particles, A and B, each of rest energy 1.40 × 102 MeV. The particles fly off in opposite directions, each with speed 0.827c relative to an inertial reference frame S. -Determine the total energy of particle A.

A space ship at rest on a launch pad has a mass of 1.00 × 105 kg. How much will its energy have increased when the ship is moving at 0.600c?

A muon has rest energy 105 MeV. What is a muon's kinetic energy when its speed is 0.95c?

A spaceship leaves our solar system at a constant speed of 0.920c and travels to a point in the Andromeda galaxy. According to astronomers in an inertial reference frame on Earth, the distance to the galaxy is 2.081 × 1022 m. What distance does the crew on the ship measure on its journey?

The power capacity of all nuclear power plants in the United States is 1.01 × 105 MW. In one hour, what is the total change in the mass of the nuclear fuel at these power plants due to the energy being taken from the reactors? Assume 100% efficiency.

A particle travels at 0.60c. Determine the ratio of its kinetic energy to its rest energy.

Determine the total energy of an electron traveling at 0.98c.

How much energy is required to accelerate a golf ball of mass 0.046 kg initially at rest to a speed of 0.75c?

In the distant future, a 5.40 × 105-kg intergalactic ship leaves Earth orbit and accelerates to a constant speed of 0.92c. Determine the difference, p - p0, between the relativistic and classical momenta of the ship.

The rest energies of three subatomic particles are: Particle X: 107 MeV; Particle Y: 140 MeV; Particle Z: 0.51 MeV. Which one of the following statements is necessarily true concerning these three particles?

Calculate the ratio of the relativistic kinetic energy to the classical kinetic energy, KE_rel/KE_class, for an electron (mass = 9.109 × 10^-³¹ kg) moving with a constant speed of 0.75c.

A proton has a mass of 1.673 × 10^-²⁷ kg. If the proton is accelerated to a speed of 0.96c, what is the magnitude of the relativistic momentum of the proton?

The temperature of a 5.00-kg lead brick is increased by 525 C°. If the specific heat capacity of lead is 128 J/(kg ^$\circ$ C°) , what is the increase in the mass of the lead brick when it has reached its final temperature?

An electron gun inside a computer monitor sends an electron toward the screen at a speed of 1.20 × 10⁸ m/s. If the mass of the electron is 9.109 × 10^-³¹ kg, what is the magnitude of its relativistic momentum?

A subatomic particle X spontaneously decays into two particles, A and B, each of rest energy 1.40 × 10² MeV. The particles fly off in opposite directions, each with speed 0.827c relative to an inertial reference frame S. -Determine the total energy of particle A.

A space ship at rest on a launch pad has a mass of 1.00 × 10⁵ kg. How much will its energy have increased when the ship is moving at 0.600c?

A spaceship leaves our solar system at a constant speed of 0.920c and travels to a point in the Andromeda galaxy. According to astronomers in an inertial reference frame on Earth, the distance to the galaxy is 2.081 × 10²² m. What distance does the crew on the ship measure on its journey?

The power capacity of all nuclear power plants in the United States is 1.01 × 10⁵ MW. In one hour, what is the total change in the mass of the nuclear fuel at these power plants due to the energy being taken from the reactors? Assume 100% efficiency.

In the distant future, a 5.40 × 10⁵-kg intergalactic ship leaves Earth orbit and accelerates to a constant speed of 0.92c. Determine the difference, p - p₀, between the relativistic and classical momenta of the ship.