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

Options

A. 37 MeV
B. 231 MeV
C. 47 MeV
D. 441 MeV

Answer

To find the kinetic energy of a muon moving at a speed of 0.95c with a rest energy of 105 MeV, we can follow these steps:

### Step 1: Understand the relationship between rest energy and rest mass
The rest energy \( E_0 \) of the muon is given by the equation:
\[ E_0 = m_0 c^2 \]
where \( m_0 \) is the rest mass and \( c \) is the speed of light. Given that \( E_0 = 105 \text{ MeV} \), we can express the rest mass as:
\[ m_0 = \frac{E_0}{c^2} \]

### Step 2: Calculate the relativistic mass
When the muon is moving at speed \( v = 0.95c \), its relativistic mass \( m \) can be calculated using the formula:
\[ m = \frac{m_0}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \]
Substituting \( v = 0.95c \):
\[ m = \frac{m_0}{\sqrt{1 - (0.95)^2}} \]

### Step 3: Calculate the kinetic energy
The total energy \( E \) of the muon when it is moving is given by:
\[ E = mc^2 \]
The kinetic energy \( KE \) can be found using the difference between the total energy and the rest energy:
\[ KE = E - E_0 = mc^2 - m_0c^2 \]
This can be rewritten as:
\[ KE = (m - m_0)c^2 \]

### Step 4: Substitute the expressions
Now we can substitute \( m \) from Step 2 into the kinetic energy equation:
\[ KE = \left(\frac{m_0}{\sqrt{1 - (0.95)^2}} - m_0\right)c^2 \]
Factoring out \( m_0 \):
\[ KE = m_0 c^2 \left(\frac{1}{\sqrt{1 - (0.95)^2}} - 1\right) \]

### Step 5: Calculate the numerical values
First, we need to calculate \( \sqrt{1 - (0.95)^2} \):
\[ \sqrt{1 - (0.95)^2} = \sqrt{1 - 0.9025} = \sqrt{0.0975} \approx 0.312 \]
Now, substituting this back into the kinetic energy equation:
\[ KE = m_0 c^2 \left(\frac{1}{0.312} - 1\right) \]
Calculating \( \frac{1}{0.312} \):
\[ \frac{1}{0.312} \approx 3.205 \]
Thus:
\[ KE = m_0 c^2 (3.205 - 1) = m_0 c^2 \cdot 2.205 \]

### Step 6: Find \( m_0 c^2 \)
Since \( m_0 c^2 = 105 \text{ MeV} \):
\[ KE = 105 \text{ MeV} \cdot 2.205 \approx 231.525 \text{ MeV} \]

### Final Answer
The kinetic energy of the muon when its speed is 0.95c is approximately:
\[ KE \approx 231.5 \text{ MeV} \]

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