An electron and a proton are detected in a cosmic ray experiment, the first with kinetic energy 10 keV, and the second with 100 keV. The ratio of their speeds is
(where `m_e` and `m_p` are masses of electron and proton respectively)

Options

A. `sqrt(1/10 m_e/m_p)`
B. `sqrt(1/10 m_p/m_e)`
C. `1/10 m_e/m_p`
D. `1/10 m_p/m_e`

Explanation

To find the ratio of the speeds of an electron and a proton based on their kinetic energies, we can follow these steps:

### Step 1: Write the formula for kinetic energy
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass and \( v \) is the speed of the object.

### Step 2: Express the speed in terms of kinetic energy
Rearranging the kinetic energy formula to solve for speed \( v \):
\[ v = \sqrt{\frac{2 \cdot KE}{m}} \]

### Step 3: Calculate the speed of the electron
Given that the kinetic energy of the electron (\( KE_e \)) is 10 keV, we can substitute this into our speed formula:
\[ v_e = \sqrt{\frac{2 \cdot KE_e}{m_e}} = \sqrt{\frac{2 \cdot 10 \, \text{keV}}{m_e}} \]

### Step 4: Calculate the speed of the proton
Similarly, for the proton, given that the kinetic energy (\( KE_p \)) is 100 keV:
\[ v_p = \sqrt{\frac{2 \cdot KE_p}{m_p}} = \sqrt{\frac{2 \cdot 100 \, \text{keV}}{m_p}} \]

### Step 5: Find the ratio of the speeds
Now, we can find the ratio of the speeds of the electron to the proton:
\[ \frac{v_e}{v_p} = \frac{\sqrt{\frac{2 \cdot 10 \, \text{keV}}{m_e}}}{\sqrt{\frac{2 \cdot 100 \, \text{keV}}{m_p}}} \]

### Step 6: Simplify the ratio
This can be simplified as follows:
\[ \frac{v_e}{v_p} = \sqrt{\frac{2 \cdot 10 \, \text{keV}}{m_e}} \cdot \sqrt{\frac{m_p}{2 \cdot 100 \, \text{keV}}} \]
\[ = \sqrt{\frac{10 \cdot m_p}{100 \cdot m_e}} = \sqrt{\frac{1}{10} \cdot \frac{m_p}{m_e}} \]

### Step 7: Final expression for the ratio
Thus, the final expression for the ratio of their speeds is:
\[ \frac{v_e}{v_p} = \sqrt{\frac{m_p}{10 \cdot m_e}} \]

### Summary
The ratio of the speeds of the electron and proton is:
\[ \frac{v_e}{v_p} = \sqrt{\frac{m_p}{10 \cdot m_e}} \]