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Chapter 29: Problem 56

If a neutron moves with a speed of \(0.99 \mathrm{c},\) what are its (a) total energy, (b) rest energy, and (c) kinetic energy?

Short Answer

Expert verified
a) Total energy: \(1.06 \times 10^{-9}\) J; b) Rest energy: \(1.51 \times 10^{-10}\) J; c) Kinetic energy: \(9.08 \times 10^{-10}\) J.

Step by step solution

01

Identify the problem

We need to calculate the total energy, rest energy, and kinetic energy of a neutron moving at a speed of \(0.99c\). The rest mass of a neutron \(m\) is approximately \(1.675 \times 10^{-27} \text{ kg}\). Additionally, we will use the speed of light \(c = 3 \times 10^8 \text{ m/s.}\) For calculations, remember the formulas: total energy \(E = \gamma mc^2\), rest energy \(E_0 = mc^2\), and kinetic energy \(K = E - E_0.\)
02

Calculate the Lorentz factor (\(\gamma\))

The Lorentz factor \(\gamma\) is calculated using the formula: \[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]Substitute \(v = 0.99c\):\[ \gamma = \frac{1}{\sqrt{1 - (0.99)^2}} = \frac{1}{\sqrt{1 - 0.9801}} = \frac{1}{\sqrt{0.0199}} \approx 7.09 \]
03

Calculate the total energy (\(E\))

Using the Lorentz factor \(\gamma\), the total energy \(E\) is given by:\[ E = \gamma mc^2 \]Substitute \(\gamma \approx 7.09\), \(m = 1.675 \times 10^{-27} \text{ kg}\), and \(c = 3 \times 10^8 \text{ m/s}\):\[ E \approx 7.09 \times 1.675 \times 10^{-27} \times (3 \times 10^8)^2 \approx 1.06 \times 10^{-9} \text{ Joules} \]
04

Calculate the rest energy (\(E_0\))

The rest energy \(E_0\) is calculated using the formula:\[ E_0 = mc^2 \]Plug in \(m = 1.675 \times 10^{-27} \text{ kg}\) and \(c = 3 \times 10^8 \text{ m/s}\):\[ E_0 = 1.675 \times 10^{-27} \times (3 \times 10^8)^2 \approx 1.51 \times 10^{-10} \text{ Joules} \]
05

Calculate the kinetic energy (\(K\))

The kinetic energy \(K\) can be found by subtracting the rest energy from the total energy:\[ K = E - E_0 \]Substituting the values obtained:\[ K \approx 1.06 \times 10^{-9} - 1.51 \times 10^{-10} \approx 9.08 \times 10^{-10} \text{ Joules} \]
06

Verify Units and Logical Consistency

Ensure that all energy values are in Joules and check that the kinetic energy is less than the total energy, as expected for a moving object.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Lorentz_factor
In the world of special relativity, the Lorentz factor, often denoted as \( \gamma \), plays a crucial role. It helps describe how time, length, and relativistic mass change for an object moving close to the speed of light. Here's how it works:

  • The Lorentz factor formula is: \[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]
  • "\( v \)" is the velocity of the object, and "\( c \)" is the speed of light.
  • As "\( v\)" approaches "\( c\)", the factor "\( \gamma\)" increases significantly, which affects the perceived time and length for the moving object.
The Lorentz factor is fundamental in computing energies of particles moving at high speeds. For example, a neutron moving at \(0.99c\) has \( \gamma \approx 7.09 \). This indicates significant relativistic effects should be considered.
rest_energy
Rest energy is another important concept in relativity. It represents the energy stored in an object due to its mass when it is at rest.

  • The formula is given by \( E_0 = mc^2 \).
  • "\( m\)" is the rest mass of the object, and "\( c\)" is the speed of light.
  • This equation, famously derived by Albert Einstein, emphasizes that mass can be seen as a concentrated form of energy.
For a neutron with a mass of approximately \(1.675 \times 10^{-27}\) kg, the rest energy calculated using this formula is around \(1.51 \times 10^{-10}\) Joules. This energy forms the baseline from which we calculate other forms of energy.
kinetic_energy
In relativistic physics, kinetic energy is a form of energy that a body possesses due to its motion. It differs from classical kinetic energy in that it factors in the relativistic effects due to high speeds.

  • The relativistic kinetic energy is computed by: \[ K = E - E_0 \]
  • "\( E\)" is the total energy, while "\( E_0\)" is the rest energy.
  • This energy highlights additional energy the object has because it's moving and not just existing.
In the case of a neutron moving at \(0.99c\), its kinetic energy is calculated to be around \(9.08 \times 10^{-10}\) Joules. It's important to note that the kinetic energy is a result of subtracting the rest energy from the total energy, showing the extra energy the neutron has due to its motion. This ensures all energy calculations remain consistent and complete.

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Most popular questions from this chapter

A small star of mass \(m\) orbits a supermassive black hole of mass \(M .\) (a) Find the orbital speed of the star if its orbital radius is \(2 R,\) where \(R\) is the Schwarzschild radius (Equation \(29-10\) ). (b) Repeat part (a) for an orbital radius equal to \(R\).

When a proton encounters an antiproton, the two particles annihilate each other, producing two gamma rays. Assuming the particles were at rest when they annihilated, find the energy of each of the two gamma rays produced. (Note: The rest energies of an antiproton and a proton are identical.)

A space probe with a rest mass of \(8.2 \times 10^{7} \mathrm{kg}\) and a speed of \(0.50 \mathrm{c}\) smashes into an asteroid at rest and becomes embedded within it. If the speed of the probe-asteroid system is \(0.26 \mathrm{c}\) after the collision, what is the rest mass of the asteroid?

As measured in Earth's frame of reference, two planets are \(424,000 \mathrm{km}\) apart. A spaceship flies from one planet to the other with a constant velocity, and the clocks on the ship show that the trip lasts only 1.00 s. How fast is the ship traveling?

When a certain capacitor is charged, its mass increases by \(8.3 \times 10^{-16} \mathrm{kg} .\) How much energy is stored in the capacitor?
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