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

Options

A. m
B. `sqrtm`
C. `(1)/(sqrtm)`
D. independent m

Answer

To solve the problem, we need to determine the relationship between the kinetic energy (KE) acquired by a mass \( m \) traveling a distance \( s \) under a constant force, starting from rest.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- The mass \( m \) starts from rest, which means the initial velocity \( u = 0 \).

2. **Using the Kinematic Equation**:
- We can use the kinematic equation to relate the final velocity \( v \) to the distance \( s \) traveled under constant acceleration \( a \):
\[ v^2 = u^2 + 2as \]
- Since \( u = 0 \), this simplifies to:
\[ v^2 = 2as \]

3. **Relating Acceleration to Force**:
- According to Newton's second law, the acceleration \( a \) can be expressed as:
\[ a = \frac{F}{m} \]
- Here, \( F \) is the constant force acting on the mass.

4. **Substituting Acceleration into the Velocity Equation**:
- Substitute \( a \) into the equation for \( v^2 \):
\[ v^2 = 2 \left(\frac{F}{m}\right) s \]

5. **Calculating Kinetic Energy**:
- The kinetic energy \( KE \) is given by:
\[ KE = \frac{1}{2} mv^2 \]
- Substitute \( v^2 \) from the previous step into the kinetic energy formula:
\[ KE = \frac{1}{2} m \left(2 \frac{F}{m} s\right) \]
- Simplifying this gives:
\[ KE = \frac{1}{2} \cdot 2 \cdot F \cdot s = F \cdot s \]

6. **Conclusion**:
- From the final expression, we see that the kinetic energy \( KE \) is directly proportional to the force \( F \) and the distance \( s \), but it is independent of the mass \( m \).

### Final Answer:
The kinetic energy acquired by a mass \( m \) in traveling a certain distance \( s \) under the action of a constant force is directly proportional to the distance \( s \) and the force \( F \), but not to the mass \( m \).