The `K.E.` acquired by a mass `m` in travelling a certain distance `d`, starting from rest, under the action of a constant force is directly propotional to

Options

A. Directly proportional to `sqrtm`
B. Directly proportional to m
C. Directly proportional to `(1)/(m)`
D. None of these

Answer

To solve the problem, we need to analyze the relationship between the kinetic energy (K.E.) acquired by a mass \( m \) traveling a distance \( d \) under the influence of a constant force. We'll derive the expression step by step.

### Step 1: Understand the scenario
A mass \( m \) starts from rest and is acted upon by a constant force \( F \). The initial velocity \( u = 0 \).

### Step 2: Use the equations of motion
According to the equations of motion, the final velocity \( v \) of the mass after traveling a distance \( d \) under constant acceleration \( a \) can be expressed as:
\[ v^2 = u^2 + 2ad \]
Since \( u = 0 \), this simplifies to:
\[ v^2 = 2ad \]

### Step 3: Relate acceleration to force
The acceleration \( a \) can be expressed in terms of the force \( F \) and mass \( m \) using Newton's second law:
\[ a = \frac{F}{m} \]
Substituting this into the equation for \( v^2 \):
\[ v^2 = 2 \left(\frac{F}{m}\right) d \]

### Step 4: Express kinetic energy
The kinetic energy \( K.E. \) acquired by the mass is given by:
\[ K.E. = \frac{1}{2} mv^2 \]
Substituting the expression for \( v^2 \) from the previous step:
\[ K.E. = \frac{1}{2} m \left(2 \frac{F}{m} d\right) \]
This simplifies to:
\[ K.E. = Fd \]

### Step 5: Analyze the relationship
From the derived expression \( K.E. = Fd \), we can see that the kinetic energy acquired by the mass is directly proportional to the force \( F \) and the distance \( d \) traveled.

### Conclusion
Thus, the kinetic energy acquired by a mass \( m \) in traveling a distance \( d \) under the action of a constant force is directly proportional to the work done, which is \( Fd \).

### Final Answer
The kinetic energy acquired by a mass \( m \) in traveling a certain distance \( d \), starting from rest, under the action of a constant force is directly proportional to \( Fd \).

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