Problem 33 - University Physics with Modern Physics

Question

A proton (rest mass \(1.67 \times 10^{ \times 27} \mathrm{kg} )\) has total energy that is 4.00 times its rest energy. What are (a) the kinetic energy of the proton; (b) the magnitude of the momentum of the proton; (c) the speed of the proton?

Step-by-Step Solution

Answer

Kinetic Energy: 3.41 MeV; Momentum: 1.27 MeV/c; Speed: 0.94c.

Step 1: Identify the Given and Required Values

We are given that the total energy of the proton is 4.00 times its rest energy. We need to find the kinetic energy, the magnitude of momentum, and the speed of the proton.

Step 2: Calculate the Rest Energy

The rest energy of the proton is calculated using the formula: \(E_0 = m_0 c^2\). Here, \(m_0 = 1.67 \times 10^{-27}\, \text{kg}\) and \(c = 3.00 \times 10^8 \, \text{m/s}\). Thus, \(E_0 = 1.67 \times 10^{-27} \times (3.00 \times 10^8)^2 \).

Step 3: Determine the Total Energy

We know the total energy \(E_{total}\) is 4 times the rest energy: \(E_{total} = 4E_0\). We use the rest energy calculated in Step 2 to determine \(E_{total}\).

Step 4: Find the Kinetic Energy

The kinetic energy \(E_k\) is given by the difference of the total energy and the rest energy: \(E_k = E_{total} - E_0\). We substitute the values obtained in Steps 2 and 3.

Step 5: Calculate the Relativistic Momentum

Momentum is given by \(p = \sqrt{(E_{total}/c)^2 - (m_0 c)^2}\). Use the expressions for \(E_{total}\) from Step 3 and the rest energy from Step 2.

Step 6: Determine the Speed of the Proton

The relativistic relation for velocity \(v\) is using the formula for relativistic kinetic energy: \(v = c \sqrt{1 - (m_0 c^2 / E_{total})^2}\). Substitute the found values to calculate \(v\).

Key Concepts

Kinetic Energy

Kinetic energy in relativistic mechanics is slightly different from classical mechanics. Instead of using just mass and velocity, we also consider the rest energy of a particle. For a proton with a given total energy that is 4 times its rest energy, the kinetic energy is derived by identifying the excess over rest energy. It is given by the formula:

• \(E_k = E_{total} - E_0\)

Here, \(E_0\) represents the rest energy, which is calculated using the proton's rest mass and the speed of light squared: \(E_0 = m_0 c^2\). Calculating this provides a clearer view of how energy is distributed between potential and kinetic forms in the relativistic context.

Momentum

In relativity, momentum is also modified from its classical form. It takes into account the relativistic effects at higher speeds. For a proton, the relativistic momentum \(p\) can be determined using the formula:

• \(p = \sqrt{{(E_{total}/c)}^2 - {(m_0 c)}^2}\)

This means that momentum is not just mass times velocity, but a quantity influenced by the total energy and the rest mass energy. The square root term ensures that as velocities approach the speed of light, the momentum increases in a way consistent with Einstein's theories, capturing the essence of mass-energy equivalence.

Speed of Light

The speed of light \(c\) is a fundamental constant in physics, valued at approximately \(3.00 \times 10^8 \, \text{m/s}\). It forms the basis of Einstein's theory of relativity, implying that no object with mass can reach or exceed this speed. This constant bridges the relationship between energy and mass, demonstrated by \(E = mc^2\). In calculations involving a proton's speed, \(c\) is used in determining kinetic energy and even the speed itself in relativistic contexts, such as with the formula for velocity:

• \(v = c \sqrt{1 - (m_0 c^2 / E_{total})^2}\)

Through this equation, it's evident that as energy increases and approaches the speed of light, relativistic effects become significant, demanding precise calculations.

Proton

Protons are subatomic particles found within an atomic nucleus, characterized by a positive electric charge and a relatively large mass compared to other particles like electrons. The mass of a proton is typically given as \(1.67 \times 10^{-27} \, \text{kg}\). In relativistic physics, the properties of protons are examined in high-energy physics experiments and scenarios, where their speeds approach a significant fraction of the speed of light.

• Rest mass energy: Use formula \(E_0 = m_0 c^2\).
• Impact in collisions: Often in particle accelerators.

Understanding the behavior of protons in such contexts is crucial for various applications, including nuclear energy production and particle physics research.