Q.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

Concept: Power–Time Relation under Constant Acceleration

Power is the rate at which work is done, $P = \vec{F} \cdot \vec{v}$. For one-dimensional motion this becomes $P = Fv$.

Under constant acceleration $a$ starting from rest, the velocity grows linearly with time:

The force is constant (Newton's second law: $F = ma$), so the power becomes …

The question asks how power scales with time when a body accelerates uniformly from rest. To see this, we need to connect power to the kinematic quantities we know.

Power is the rate at which work is done, or equivalently, the rate at which energy is delivered to the body. The instantaneous power delivered by a force is given by

For one-dimensional motion, this simplifies to $P = Fv$. Now we need to figure out how $F$ and $v$ each depend on time under constant acceleration.

Step-by-step reasoning

1. Identify the force. The body has constant acceleration $a$. By Newton's second law, the net force is

Since both mass and acceleration are constant, the force is constant throughout the motion.

2. Find the velocity as a function of time. The body starts from rest, so $v_0 = 0$. Under constant acceleration, the velocity at time $t$ is

So velocity grows linearly with time.

3. Express power in terms of time. Substituting into the power formula:

Since $m$ and $a$ are constants, we have

$P(t) \propto t$ …

Concept: Power as the Rate of Work — $P = Fv$

Step 1: Find the force under constant acceleration.

By Newton's second law, $F = ma$, constant throughout the motion.

Step 2: Find velocity as a function of time, starting from rest.

Step 3: Substitute into $P = Fv$. …