A force of magnitude `F_1` , accelerates a body of mass m from rest to a speed v. Another force `F_2` accelerates a body of mass 2m from rest to a speed 2v. The ratio of work done by force `F_2` to that by force `F_1` is

Options

A. `1:16`
B. `16:1`
C. `1:8`
D. `8:1`

Answer

To solve the problem, we need to find the ratio of work done by force \( F_2 \) to that done by force \( F_1 \). Let's break this down step by step.

### Step 1: Calculate Work Done by Force \( F_1 \)

1. **Identify the parameters**:
- Mass of the first body, \( m \).
- Final speed, \( v \).
- Initial speed, \( u = 0 \) (starts from rest).

2. **Calculate acceleration \( A_1 \)**:
\[ A_1 = \frac{F_1}{m} \]

3. **Use the third equation of motion**:
\[ v^2 = u^2 + 2A_1 S_1 \]
Substituting \( u = 0 \):
\[ v^2 = 2A_1 S_1 \implies S_1 = \frac{v^2}{2A_1} \]

4. **Calculate work done \( W_1 \)**:
\[ W_1 = F_1 \cdot S_1 = F_1 \cdot \frac{v^2}{2A_1} \]
Substitute \( A_1 = \frac{F_1}{m} \):
\[ W_1 = F_1 \cdot \frac{v^2}{2 \cdot \frac{F_1}{m}} = \frac{mv^2}{2} \]

### Step 2: Calculate Work Done by Force \( F_2 \)

1. **Identify the parameters**:
- Mass of the second body, \( 2m \).
- Final speed, \( 2v \).
- Initial speed, \( u = 0 \) (starts from rest).

2. **Calculate acceleration \( A_2 \)**:
\[ A_2 = \frac{F_2}{2m} \]

3. **Use the third equation of motion**:
\[ (2v)^2 = 0 + 2A_2 S_2 \implies 4v^2 = 2A_2 S_2 \implies S_2 = \frac{4v^2}{2A_2} = \frac{2v^2}{A_2} \]

4. **Calculate work done \( W_2 \)**:
\[ W_2 = F_2 \cdot S_2 = F_2 \cdot \frac{2v^2}{A_2} \]
Substitute \( A_2 = \frac{F_2}{2m} \):
\[ W_2 = F_2 \cdot \frac{2v^2}{\frac{F_2}{2m}} = F_2 \cdot \frac{4mv^2}{F_2} = 4mv^2 \]

### Step 3: Find the Ratio of Work Done

Now, we can find the ratio of work done by \( F_2 \) to that done by \( F_1 \):
\[ \frac{W_2}{W_1} = \frac{4mv^2}{\frac{mv^2}{2}} = \frac{4mv^2 \cdot 2}{mv^2} = 8 \]

### Final Answer
The ratio of work done by force \( F_2 \) to that by force \( F_1 \) is:
\[ \frac{W_2}{W_1} = 8:1 \]