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 determine how the velocity acquired by a mass \( m \) traveling a distance \( d \) under a constant force relates to these variables. Here’s a step-by-step breakdown:

### Step 1: Understand the Given Information
We have:
- A mass \( m \) starting from rest (initial velocity \( u = 0 \)).
- It is acted upon by a constant force \( F \).
- The distance traveled is \( d \).

### Step 2: Determine the Acceleration
Using Newton's second law, the acceleration \( a \) of the mass can be expressed as:
\[ a = \frac{F}{m} \]

### Step 3: Use the Third Equation of Motion
The third equation of motion relates the final velocity \( v \), initial velocity \( u \), acceleration \( a \), and distance \( d \):
\[ v^2 = u^2 + 2ad \]
Since the initial velocity \( u = 0 \), this simplifies to:
\[ v^2 = 2ad \]

### Step 4: Substitute for Acceleration
Now, substitute \( a \) from Step 2 into the equation:
\[ v^2 = 2 \left(\frac{F}{m}\right) d \]
This can be rearranged to:
\[ v^2 = \frac{2Fd}{m} \]

### Step 5: Solve for Velocity
Taking the square root of both sides gives us the expression for velocity:
\[ v = \sqrt{\frac{2Fd}{m}} \]

### Step 6: Identify Proportional Relationships
From the equation \( v = \sqrt{\frac{2Fd}{m}} \), we can see that:
- \( v \) is directly proportional to \( \sqrt{F} \) (the force).
- \( v \) is directly proportional to \( \sqrt{d} \) (the distance).
- \( v \) is inversely proportional to \( \sqrt{m} \) (the mass).

### Conclusion
Thus, the velocity acquired by the mass \( m \) in traveling a distance \( d \) under a constant force \( F \) is directly proportional to \( \sqrt{F} \) and \( \sqrt{d} \), and inversely proportional to \( \sqrt{m} \).

### Final Answer
The velocity acquired is directly proportional to \( \sqrt{F} \) and \( \sqrt{d} \) and inversely proportional to \( \sqrt{m} \).

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