Question 1

Assume that a certain city consumes electrical energy at an average rate of 2.0 × 10⁹ W.What would be the mass change in producing enough energy to keep this city running for 21 weeks? (c = 3.00 × 10⁸ m/s)

Accepted Answer

To find the mass change, we can use the equation derived from Einstein's theory of relativity, $E = mc^2$, where $E$ is the energy, $m$ is the mass, and $c$ is the speed of light. First, we need to calculate the total energy consumed by the city over 21 weeks.Given:- Average power consumption = $2.0 \times 10^{19}$ W (Watts are Joules per second)- Time = 21 weeks = $21 \times 7 \times 24 \times 60 \times 60$ seconds (converting weeks to seconds)- Speed of light, $c = 3.00 \times 10^8$ m/sThe total energy consumed, $E$, can be calculated as power multiplied by time:$$E = P \times t = 2.0 \times 10^{19} \times (21 \times 7 \times 24 \times 60 \times 60)$$To find the mass change, $m$, we rearrange the equation $E = mc^2$ to $m = \frac{E}{c^2}$.Substituting the values:$$m = \frac{E}{(3.00 \times 10^8)^2}$$After calculating $E$ with the given values and then calculating $m$, we find that the mass change is approximately 0.28 kg, which corresponds to option A. This calculation demonstrates the principle of mass-energy equivalence, showing how a relatively small mass change can produce a significant amount of energy.

Question 2

A relativistic proton has a momentum of 1.0 × 10^-17kg ∙ m/s and a rest energy of 0.15 nJ.What is the kinetic energy of this proton? (c = 3.00 × 10⁸ m/s,m_proton = 1.67 × 10^-27 kg)

Accepted Answer

The answer of A relativistic proton has a momentum of...

Question 3

During a nuclear reaction,1.7 × 10^-4 J of energy is released. What is the resulting change in mass of the particles involved? (c = 3.00 × 10⁸ m/s)

Accepted Answer

The answer of During a nuclear reaction,1.7 × 10^-4 J...

Question 4

An electron is accelerated from rest through a potential difference of 50.0 kV.What is the TOTAL energy of the electron? (c = 3.00 × 10⁸ m/s,e = 1.60 × 10^-19 C,m_el = 9.11 × 10^-31 kg)

Accepted Answer

The total energy of the electron is the sum of its kinetic energy and potential energy. Kinetic energy = (1/2)mv^2 where m is the mass of the electron and v is its velocity. Potential energy = qV where q is the charge of the electron and V is the potential difference. Initially, the electron is at rest so its kinetic energy is zero. Therefore, the total energy of the electron is equal to its potential energy, which is given by (1.60 × 10^-19 C)(50.0 × 10^3 V) = 8.00 × 10^-14 J.

Question 5

A proton in a certain particle accelerator has a kinetic energy that is equal to its rest energy.What is the momentum of the proton as measured by a physicist working with the accelerator? (c = 3.00 × 10⁸ m/s,m_proton = 1.67 × 10^-27 kg)

Accepted Answer

The answer of A proton in a certain particle accelerator...

Question 6

During a nuclear reaction,the particles involved lose 4.8 × 10^-28kg of mass. How many joules of energy are released by this reaction? (c = 3.00 × 10⁸ m/s)

Accepted Answer

The energy released can be calculated using Einstein's equation, E=mc^2, where m is the mass lost (4.8 × 10^-28 kg) and c is the speed of light (3.00 × 10^8 m/s). Plugging in the values, E = (4.8 × 10^-28 kg)(3.00 × 10^8 m/s)^2 = 4.32 × 10^-11 J, which rounds to 4.3 × 10^-11 J.

Question 7

A proton in a certain particle accelerator has a kinetic energy that is equal to its rest energy.What is the TOTAL energy of the proton as measured by a physicist working with the accelerator? (c = 3.00 × 10⁸ m/s,m_proton = 1.67 × 10^-27 kg)

Accepted Answer

The answer of A proton in a certain particle accelerator...

Question 8

As a spaceship is moving toward Earth,an Earthling measures its length to be 325 m,while the captain on board radios that her spaceship's length is 1150 m.(c = 3.00 × 10⁸ m/s) (a)How fast is the rocket moving relative to Earth? (b)What is the TOTAL energy of a 75.0-kg crewman as measured by (i)the captain in the rocket and (ii)the Earthling?

Accepted Answer

The answer of As a spaceship is moving toward Earth,an...

Question 9

How fast must a proton move so that its kinetic energy is 80% of its total energy?

Accepted Answer

The total energy of a particle is given by the relativistic equation E = γmc², where γ is the Lorentz factor and m is the rest mass of the particle. The kinetic energy of a particle is given by the classical equation KE = ½mv². Setting KE = 0.8E, we have
0.8E = γmc² + mc²
0.8E = (γ + 1)mc²
γ + 1 = 0.8E/mc²
γ = 0.8E/mc² - 1
To find the velocity v of the proton, we use the equation γ = 1/√(1-v²/c²). Substituting the expression for γ, we get
0.8E/mc² - 1 = 1/√(1-v²/c²)
Solving for v/c, we get v/c = √[1 - (0.8mc²/E)²]. Substituting the mass and energy of the proton, we get v/c = √[1 - (0.8*1.67x10^-27*3x10^8*3x10^8/(938.3x10^6*1.602x10^-19))^2] ≈ 0.98. Therefore, the best choice is D) 0.98c.

Quiz 14: Special Relativity

Assume that a certain city consumes electrical energy at an average rate of 2.0 × 109 W.What would be the mass change in producing enough energy to keep this city running for 21 weeks? (c = 3.00 × 108 m/s)

A relativistic proton has a momentum of 1.0 × 10-17 kg ∙ m/s and a rest energy of 0.15 nJ.What is the kinetic energy of this proton? (c = 3.00 × 108 m/s,mproton = 1.67 × 10-27 kg)

During a nuclear reaction,1.7 × 10-4 J of energy is released. What is the resulting change in mass of the particles involved? (c = 3.00 × 108 m/s)

An electron is accelerated from rest through a potential difference of 50.0 kV.What is the TOTAL energy of the electron? (c = 3.00 × 108 m/s,e = 1.60 × 10-19 C,mel = 9.11 × 10-31 kg)

A proton in a certain particle accelerator has a kinetic energy that is equal to its rest energy.What is the momentum of the proton as measured by a physicist working with the accelerator? (c = 3.00 × 108 m/s,mproton = 1.67 × 10-27 kg)

During a nuclear reaction,the particles involved lose 4.8 × 10-28 kg of mass. How many joules of energy are released by this reaction? (c = 3.00 × 108 m/s)

A proton in a certain particle accelerator has a kinetic energy that is equal to its rest energy.What is the TOTAL energy of the proton as measured by a physicist working with the accelerator? (c = 3.00 × 108 m/s,mproton = 1.67 × 10-27 kg)