The velocity acquired by a mass m in travelling a certain distance d starting from rest under the action of a constant force is directly proportional to :-

Options

A. `sqrt(m)`
B. Independent of m
C. `(1)/sqrt(m)`
D. m

Explanation

To solve the problem, we need to analyze the relationship between the velocity acquired by a mass \( m \) while traveling a distance \( d \) under the influence of a constant force. We will derive the expression step by step.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Mass of the object: \( m \)
- Initial velocity: \( u = 0 \) (starting from rest)
- Final velocity: \( v \)
- Distance traveled: \( s = d \)
- Force acting on the object: \( F \) (constant)

2. **Use Newton's Second Law:**
According to Newton's second law, the force acting on an object is equal to the mass of the object multiplied by its acceleration:
\[ F = m \cdot a \quad \Rightarrow \quad a = \frac{F}{m} \]
where \( a \) is the constant acceleration.

3. **Use the Kinematic Equation:**
Since the object starts from rest, we can use the kinematic equation that relates final velocity, initial velocity, acceleration, and distance:
\[ v^2 = u^2 + 2as \]
Substituting \( u = 0 \) and \( s = d \):
\[ v^2 = 0 + 2ad \quad \Rightarrow \quad v^2 = 2ad \]

4. **Substitute the Expression for Acceleration:**
Now, substitute \( a = \frac{F}{m} \) into the equation:
\[ v^2 = 2 \left(\frac{F}{m}\right) d \]
Simplifying this gives:
\[ v^2 = \frac{2Fd}{m} \]

5. **Express Velocity in Terms of Mass:**
To find the relationship of \( v \) with respect to \( m \), we take the square root of both sides:
\[ v = \sqrt{\frac{2Fd}{m}} \]

6. **Identify the Proportionality:**
From the equation \( v = \sqrt{\frac{2Fd}{m}} \), we can see that:
\[ v \propto \frac{1}{\sqrt{m}} \]
This means that the velocity acquired by the mass is inversely proportional to the square root of the mass.

### Conclusion:
The velocity acquired by a mass \( m \) in traveling a distance \( d \) starting from rest under the action of a constant force is directly proportional to \( \frac{1}{\sqrt{m}} \).