A body is initially at rest. It undergoes one dimensional motion with constant acceleration. The power delivered to it at time t is proportional to

Options

A. `t^(1//2)`
B. `t`
C. `t^(3//2)`
D. `t^2`

Explanation

To solve the problem of determining how the power delivered to a body undergoing one-dimensional motion with constant acceleration is proportional to time \( t \), we can follow these steps:

### Step 1: Understand the Initial Conditions
The body is initially at rest, which means its initial velocity \( u = 0 \).

### Step 2: Write the Equation for Velocity
Since the body is moving with constant acceleration \( a \), we can use the equation of motion to find the velocity \( v \) at time \( t \):
\[ v = u + at \]
Substituting \( u = 0 \):
\[ v = at \]

### Step 3: Write the Expression for Force
The force \( F \) acting on the body can be expressed using Newton's second law:
\[ F = ma \]
where \( m \) is the mass of the body and \( a \) is the constant acceleration.

### Step 4: Write the Expression for Power
Power \( P \) is defined as the rate at which work is done or the product of force and velocity:
\[ P = F \cdot v \]
Substituting the expressions for force and velocity:
\[ P = (ma)(at) = ma^2t \]

### Step 5: Determine the Proportionality
From the expression \( P = ma^2t \), we can see that the power \( P \) is directly proportional to time \( t \) since \( m \) and \( a \) are constants. Therefore, we can conclude:
\[ P \propto t \]

### Conclusion
The power delivered to the body at time \( t \) is proportional to \( t \).