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

Options

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

Answer

To solve the problem, we will use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- A constant force \( F \) is applied to a stationary particle of mass \( m \).
- Initial velocity \( u = 0 \) (since the particle is stationary).

2. **Apply Newton's Second Law:**
According to Newton's second law:
\[ F = m \cdot a \]
where \( a \) is the acceleration of the particle.

3. **Express Acceleration in Terms of Force and Mass:**
Rearranging the equation gives:
\[ a = \frac{F}{m} \]

4. **Relate Acceleration to Change in Velocity:**
Acceleration is defined as the change in velocity over time:
\[ a = \frac{dV}{dt} \]
where \( dV \) is the change in velocity and \( dt \) is the change in time.

5. **Set the Two Expressions for Acceleration Equal:**
We can equate the two expressions for acceleration:
\[ \frac{dV}{dt} = \frac{F}{m} \]

6. **Rearranging the Equation:**
Rearranging gives:
\[ dV = \frac{F}{m} \cdot dt \]

7. **Integrate Both Sides:**
To find the velocity \( V \) after a time \( t \), we integrate both sides:
\[ \int dV = \int \frac{F}{m} dt \]
This results in:
\[ V = \frac{F}{m} t + C \]
where \( C \) is the constant of integration. Since the initial velocity \( u = 0 \), we have \( C = 0 \).

8. **Final Expression for Velocity:**
Thus, the velocity attained by the particle is:
\[ V = \frac{F}{m} t \]

9. **Determine Proportionality:**
From the equation \( V = \frac{F}{m} t \), we can see that the velocity \( V \) is directly proportional to the force \( F \) and the time \( t \), and inversely proportional to the mass \( m \).

### Conclusion:
The velocity attained by the particle in a certain interval of time will be proportional to \( \frac{F}{m} \cdot t \).