A man has twice the mass of a boy and has half the kinetic energy of the boy. The ratio of the speeds of the man and the boy must be

Options

A. `2 : 1`
B. `4 : 1`
C. `1 : 4`
D. `1 : 2`

Explanation

To solve the problem, let's denote the following:

- Mass of the boy = \( m \)
- Mass of the man = \( 2m \) (since the man has twice the mass of the boy)
- Kinetic energy of the boy = \( K_b \)
- Kinetic energy of the man = \( K_m \) (which is half of the boy's kinetic energy, so \( K_m = \frac{1}{2} K_b \))

The formula for kinetic energy is given by:

\[ K = \frac{1}{2} mv^2 \]

### Step 1: Write the kinetic energy equations for both the man and the boy.

For the boy, we have:

\[ K_b = \frac{1}{2} m v_b^2 \]

For the man, we have:

\[ K_m = \frac{1}{2} (2m) v_m^2 = m v_m^2 \]

### Step 2: Set up the relationship between the kinetic energies.

Since \( K_m = \frac{1}{2} K_b \), we can substitute the expressions for kinetic energy:

\[ m v_m^2 = \frac{1}{2} \left( \frac{1}{2} m v_b^2 \right) \]

This simplifies to:

\[ m v_m^2 = \frac{1}{4} m v_b^2 \]

### Step 3: Cancel the mass \( m \) from both sides.

Assuming \( m \neq 0 \), we can divide both sides by \( m \):

\[ v_m^2 = \frac{1}{4} v_b^2 \]

### Step 4: Take the square root of both sides to find the ratio of speeds.

Taking the square root gives:

\[ v_m = \frac{1}{2} v_b \]

### Step 5: Write the ratio of the speeds.

The ratio of the speeds of the man to the boy is:

\[ \frac{v_m}{v_b} = \frac{1}{2} \]

### Final Answer:

The ratio of the speeds of the man and the boy is \( \frac{1}{2} \).

---