Two bodies of masses 1 kg and 2 kg moving withy 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 different masses but moving with the same velocity and stopped by the same force, we can follow these steps:

### Step 1: Understand the 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 velocity of the object.

### Step 2: Calculate the Kinetic Energies
For the two bodies:
- For the 1 kg mass:
\[ KE_1 = \frac{1}{2} \times 1 \times v^2 = \frac{1}{2} v^2 \]
- For the 2 kg mass:
\[ KE_2 = \frac{1}{2} \times 2 \times v^2 = v^2 \]

### Step 3: Relate Work Done to Stopping Distance
The work done by the force to stop the object is equal to the change in kinetic energy. If \( F \) is the force and \( s \) is the stopping distance, we have:
\[ F \cdot s = KE \]
Thus, for each mass:
- For the 1 kg mass:
\[ F \cdot s_1 = \frac{1}{2} v^2 \]
- For the 2 kg mass:
\[ F \cdot s_2 = v^2 \]

### Step 4: Set Up the Ratios
Now, we can set up the ratio of the stopping distances \( s_1 \) and \( s_2 \):
\[ \frac{s_1}{s_2} = \frac{KE_1}{KE_2} \]

Substituting the kinetic energies we calculated:
\[ \frac{s_1}{s_2} = \frac{\frac{1}{2} v^2}{v^2} = \frac{1/2}{1} = \frac{1}{2} \]

### Step 5: Conclusion
Thus, the ratio of the stopping distances \( s_1 \) and \( s_2 \) is:
\[ \frac{s_1}{s_2} = \frac{1}{2} \]

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

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