A body moves under the action of a constant force along a straight line. The instantaneous power developed by this force with time `t` is correctly represented by.

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining how instantaneous power developed by a constant force changes with time, we can follow these steps:

### Step 1: Understand the relationship between power, force, and velocity
The instantaneous power \( P \) developed by a force \( F \) acting on a body is given by the equation:
\[ P = F \cdot V \]
where \( V \) is the velocity of the body.

### Step 2: Analyze the motion under constant force
Since the body is moving under the action of a constant force along a straight line, we know that the force \( F \) is constant. The acceleration \( a \) of the body can be expressed as:
\[ a = \frac{F}{m} \]
where \( m \) is the mass of the body.

### Step 3: Relate velocity to time
If the body starts from rest, the velocity \( V \) of the body at time \( t \) can be expressed as:
\[ V = a \cdot t = \left(\frac{F}{m}\right) \cdot t \]

### Step 4: Substitute the expression for velocity into the power equation
Now, substituting the expression for velocity \( V \) into the power equation:
\[ P = F \cdot V = F \cdot \left(\frac{F}{m} \cdot t\right) \]
This simplifies to:
\[ P = \frac{F^2}{m} \cdot t \]

### Step 5: Analyze the relationship between power and time
From the equation \( P = \frac{F^2}{m} \cdot t \), we can see that power \( P \) is directly proportional to time \( t \). This indicates that as time increases, the power also increases linearly.

### Conclusion
The instantaneous power developed by the constant force with time \( t \) is represented by a linear function of time. Therefore, the correct option is:
**Option B**.

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