Question

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

• A. t^1/2
• B. t
• C. t^3/2
• D. t²

Answer 1

The Correct Option is B

The power delivered to a body is the rate at which work is done, and it can be expressed as: \[ P = F \cdot v \] Where:
\( P \) is the power,
\( F \) is the force,
\( v \) is the velocity.
For a body undergoing motion with constant acceleration, the velocity at any time \( t \) is given by: \[ v = at \] Where \( a \) is the constant acceleration. The force acting on the body is given by Newton's second law: \[ F = ma \] Where \( m \) is the mass of the body and \( a \) is the acceleration. Substituting the expressions for \( F \) and \( v \) into the equation for power: \[ P = ma \cdot at = ma^2 t \] Since the body starts from rest and has constant acceleration, the power delivered to the body is proportional to \( t \).
Thus, the power delivered to the body at time \( t \) is proportional to \( t \).

The correct answer is (B) : t.

Answer 2

Given that the body is initially at rest and undergoes uniform acceleration, we can use the following kinematic equations: 1. The velocity \( v \) at time \( t \) is given by: \[ v = at \] Where \( a \) is the constant acceleration. 2. The displacement \( x \) at time \( t \) is given by: \[ x = \frac{1}{2}at^2 \] Now, the power \( P \) delivered to the body is given by the formula: \[ P = F \cdot v \] Where \( F \) is the force applied to the body and \( v \) is the velocity of the body. Since \( F = ma \), where \( m \) is the mass of the body, the power becomes: \[ P = ma \cdot v = ma \cdot at = ma^2 t \] Thus, the power is proportional to \( t \), as \( P \propto t \). Therefore, the correct answer is \( t \), which corresponds to option (B).