Two bodies of masses 1 kg and 2 kg moving with same velocities are stopped by the same force. Then, the ratio of their stopping distances is

Options

A. `1:2`
B. `2:1`
C. `sqrt 2:1`
D. `1:sqrt2`

Explanation

To solve the problem of finding the ratio of stopping distances for two bodies of masses 1 kg and 2 kg moving with the same velocity and stopped by the same force, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Mass of the first body, \( m_1 = 1 \, \text{kg} \)
- Mass of the second body, \( m_2 = 2 \, \text{kg} \)
- Both bodies are moving with the same velocity, \( v \).

2. **Understand the Concept of Stopping Distance**:
- The stopping distance \( s \) can be derived from the work-energy principle, which states that the work done by the force is equal to the change in kinetic energy.
- The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]

3. **Calculate the Kinetic Energies**:
- For the first body:
\[ KE_1 = \frac{1}{2} m_1 v^2 = \frac{1}{2} (1) v^2 = \frac{1}{2} v^2 \]
- For the second body:
\[ KE_2 = \frac{1}{2} m_2 v^2 = \frac{1}{2} (2) v^2 = v^2 \]

4. **Relate Kinetic Energy to Stopping Distance**:
- According to the work-energy principle, the work done by the force to stop the body is equal to the kinetic energy:
\[ F \cdot s = KE \]
- For the first body:
\[ F \cdot s_1 = KE_1 \Rightarrow s_1 = \frac{KE_1}{F} \]
- For the second body:
\[ F \cdot s_2 = KE_2 \Rightarrow s_2 = \frac{KE_2}{F} \]

5. **Find the Ratio of Stopping Distances**:
- The ratio of stopping distances \( \frac{s_1}{s_2} \) can be expressed as:
\[ \frac{s_1}{s_2} = \frac{KE_1 / F}{KE_2 / F} = \frac{KE_1}{KE_2} \]
- Substitute the values of kinetic energies:
\[ \frac{s_1}{s_2} = \frac{\frac{1}{2} v^2}{v^2} = \frac{1/2}{1} = \frac{1}{2} \]

6. **Conclusion**:
- Therefore, the ratio of the stopping distances is:
\[ \frac{s_1}{s_2} = \frac{1}{2} \quad \text{or} \quad s_1 : s_2 = 1 : 2 \]

### Final Answer:
The ratio of their stopping distances is \( 1 : 2 \).