A constant force(F) is applied on a stationary particle of mass 'm'. The velocity attained by the particle in a certain displacement will be proportional to

Options

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

Answer

To solve the problem, we need to determine how the velocity attained by a particle of mass 'm' under the influence of a constant force 'F' is related to the displacement.

### Step-by-Step Solution:

1. **Understanding the Force and Mass Relationship**:
According to Newton's second law, the net force acting on an object is equal to the mass of the object multiplied by its acceleration:
\[ F = m \cdot a \]
where \( a \) is the acceleration of the particle.

2. **Expressing Acceleration in Terms of Velocity**:
Acceleration can be expressed as the change in velocity over time:
\[ a = \frac{dv}{dt} \]
Thus, we can rewrite the force equation as:
\[ F = m \cdot \frac{dv}{dt} \]

3. **Using the Chain Rule**:
We can express acceleration in terms of displacement \( x \) as follows:
\[ a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} \]
Here, \( \frac{dx}{dt} \) is the velocity \( v \). Therefore, we can rewrite the force equation:
\[ F = m \cdot v \cdot \frac{dv}{dx} \]

4. **Rearranging the Equation**:
Rearranging gives us:
\[ \frac{F}{m} \cdot dx = v \cdot dv \]

5. **Integrating Both Sides**:
Now, we integrate both sides. The left side integrates with respect to \( x \) and the right side with respect to \( v \):
\[ \int \frac{F}{m} \, dx = \int v \, dv \]
This results in:
\[ \frac{F}{m} \cdot x = \frac{v^2}{2} + C \]
Assuming the initial velocity is zero (the particle starts from rest), the constant \( C \) will be zero.

6. **Solving for Velocity**:
Thus, we have:
\[ \frac{F}{m} \cdot x = \frac{v^2}{2} \]
Rearranging gives:
\[ v^2 = \frac{2Fx}{m} \]
Taking the square root of both sides results in:
\[ v = \sqrt{\frac{2Fx}{m}} \]

7. **Determining Proportionality**:
From the equation \( v = \sqrt{\frac{2Fx}{m}} \), we can see that the velocity \( v \) is directly proportional to the square root of the displacement \( x \) and the force \( F \), and inversely proportional to the square root of the mass \( m \).

### Conclusion:
Thus, the velocity attained by the particle in a certain displacement is proportional to:
\[ v \propto \sqrt{\frac{F}{m}} \cdot \sqrt{x} \]