A body moves in a straight line (unidirectionally) under a source that delivers energy at a constant power. Wh

NCERT Exemplar · Q11

d

(a) A straight line through the origin (

d

proportional to

t

(b) A curve rising from the origin whose slope keeps increasing (

d

growing faster than linearly, like

t^{3/2}

(d) A curve rising from the origin whose slope keeps decreasing (concave-down, like a square-root shape).

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Concept understanding — Power Time Relation

Power and Time: The Intuition

Think about lifting a heavy box. If you lift it slowly, you feel tired but you manage. If you lift it fast — the same box to the same height — you feel a much greater strain. The work done (force × distance) is identical in both cases. So why does the fast lift feel harder?

The answer is power. Power tells you not just how much work is done, but how quickly it is done. The fast lift requires more power because the same work is compressed into a shorter time.

Note

Everyday language often confuses power with energy. A "powerful" car isn't one that uses more fuel (energy) — it's one that can accelerate faster or climb a hill at higher speed, meaning it delivers energy more quickly.

The Precise Statement

Power is defined as the rate at which work is done (or energy is transferred). Mathematically:

P = \frac{W}{t}

where:

$P$ = power (watts, W)
$W$ = work done (joules, J)
$t$ = time taken (seconds, s)

This is the power-time relation in its simplest form: power is work divided by time.

P = \frac{W}{t}

If you rearrange it, you get two other useful forms:

W = P \times t \quad \text{and} \quad t = \frac{W}{P}

These tell you:

To do a fixed amount of work, using more power means less time.
To run a device for a fixed time, more power means more work (and more energy consumed).

A Concrete Example

Suppose you need to lift a 10 kg mass to a height of 2 metres. The work done against gravity is:

W = mgh = 10 \times 9.8 \times 2 = 196 \text{ J}

Now consider two cases:

Case	Time taken	Power required
Slow lift	4 seconds	$P = \frac{196}{4} = 49 \text{ W}$
Fast lift	1 second	$P = \frac{196}{1} = 196 \text{ W}$

The fast lift requires four times the power — that's why it feels much harder, even though the work is the same.

Watch out

A common mistake is to think power and work are the same thing. They are not. Work is the total energy transferred; power is the rate of that transfer. A 100 W bulb uses 100 J of energy every second, but if you leave it on for an hour, the total work (energy used) is $100 \times 3600 = 360,000 \text{ J}$ .

Why This Matters for Exams

The power-time relation appears in two main forms in problems:

Direct calculation: Given work and time, find power (or any missing variable). …

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Constant power means $\tfrac12 mv^2 = Pt$ , so $v \propto t^{1/2}$ and displacement $d \propto t^{3/2}$ — a curve that starts at the origin and gets steeper with time. Correct option (B).

Derivation

Constant power $P$ delivered from rest. By the work–energy theorem, work done $=Pt$ equals kinetic energy:

Pt = \tfrac{1}{2}mv^2 \implies v = \sqrt{\frac{2P}{m}}\;t^{1/2}

Integrating $v = dx/dt$ :

d = \sqrt{\frac{2P}{m}}\cdot\frac{t^{3/2}}{3/2} \propto t^{3/2}

Since $d\propto t^{3/2}$ , the graph rises from the origin with a slope (velocity) that increases with time — concave up.

Why the others fail …

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Q.A body moves in a straight line (unidirectionally) under a source that delivers energy at a constant power. Which of the following displacement–time ( $d$ versus $t$ ) curves correctly represents its motion?

Concept understanding — Power Time Relation

Power and Time: The Intuition

The Precise Statement

A Concrete Example

Why This Matters for Exams

Derivation

Why the others fail …

Concept: Constant Power, Work-Energy Theorem and Kinematics

Q.A body moves in a straight line (unidirectionally) under a source that delivers energy at a constant power. Which of the following displacement–time (ddd versus ttt) curves correctly represents its motion?

Derivation

Why the others fail …

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Q.A body moves in a straight line (unidirectionally) under a source that delivers energy at a constant power. Which of the following displacement–time ( $d$ versus $t$ ) curves correctly represents its motion?

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