The kinetic energy acquired by a mass `m` travelling a certain distance d, starting from rest, under the action of a force F such that the force F is directly proportional to t is

Options

A. (a) Directly proportional to `t^2`
B. (b) Independent of t
C. (c) Directly proportional to `t^4`
D. (d) Directly proportional to `t`

Explanation

To solve the problem of finding the kinetic energy acquired by a mass \( m \) traveling a distance \( d \) under the action of a force \( F \) that is directly proportional to time \( t \), we can follow these steps:

### Step 1: Establish the relationship between force and time
Given that the force \( F \) is directly proportional to time \( t \), we can express this as:
\[ F = k t \]
where \( k \) is a constant of proportionality.

### Step 2: Relate force to acceleration
From Newton's second law, we know that force can also be expressed as:
\[ F = m \frac{dv}{dt} \]
Setting the two expressions for force equal gives:
\[ m \frac{dv}{dt} = k t \]

### Step 3: Rearrange and integrate
We can rearrange the equation to separate variables:
\[ m \, dv = k t \, dt \]
Now, we integrate both sides. The left side integrates with respect to \( v \) and the right side with respect to \( t \):
\[ \int m \, dv = \int k t \, dt \]
This gives:
\[ m v = \frac{k t^2}{2} + C \]
Since the mass starts from rest, when \( t = 0 \), \( v = 0 \), thus \( C = 0 \). Therefore:
\[ m v = \frac{k t^2}{2} \]

### Step 4: Solve for velocity \( v \)
Now we can solve for \( v \):
\[ v = \frac{k t^2}{2m} \]

### Step 5: Calculate kinetic energy
The kinetic energy \( KE \) is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
Substituting the expression for \( v \):
\[ KE = \frac{1}{2} m \left(\frac{k t^2}{2m}\right)^2 \]
This simplifies to:
\[ KE = \frac{1}{2} m \cdot \frac{k^2 t^4}{4m^2} = \frac{k^2 t^4}{8m} \]

### Step 6: Determine the relationship of kinetic energy with time
From the expression \( KE = \frac{k^2 t^4}{8m} \), we can see that kinetic energy is directly proportional to \( t^4 \):
\[ KE \propto t^4 \]

### Final Answer
Thus, the kinetic energy acquired by the mass \( m \) is directly proportional to \( t^4 \).

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