A motor drives a body along a straight line with a constant force. The power P developed by the motor must vary with time t as

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the relationship between power, force, and velocity when a motor drives a body with a constant force. Here’s a step-by-step solution:

### Step 1: Understand the relationship between force, mass, and acceleration
Given that the motor drives the body with a constant force \( F \), we can express this using Newton's second law:
\[ F = ma \]
where \( m \) is the mass of the body and \( a \) is its acceleration. Since the force is constant, the acceleration \( a \) must also be constant.

### Step 2: Relate velocity to time
Using the equation of motion for uniformly accelerated motion, we have:
\[ v = u + at \]
Assuming the initial velocity \( u = 0 \) (the body starts from rest), this simplifies to:
\[ v = at \]
This shows that the velocity \( v \) of the body increases linearly with time \( t \) since \( a \) is constant.

### Step 3: Write the expression for power
The power \( P \) developed by the motor is given by the formula:
\[ P = F \cdot v \]
Substituting the expression for velocity from Step 2:
\[ P = F \cdot (at) \]
Since \( F \) is constant, we can rewrite this as:
\[ P = (F \cdot a) t \]
This indicates that power \( P \) is directly proportional to time \( t \).

### Step 4: Conclusion about the relationship
From the equation \( P = (F \cdot a) t \), we can conclude that as time increases, power also increases linearly. Thus, we can express this relationship as:
\[ P \propto t \]

### Step 5: Identify the correct graph
Since power varies linearly with time, the graph of power \( P \) versus time \( t \) will be a straight line passing through the origin.

### Final Answer
The power \( P \) developed by the motor varies with time \( t \) as:
\[ P \propto t \]
The correct option is that the graph of power versus time is a straight line passing through the origin.

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