An object of mass 1000 g experiences a time dependent force $\vec{F}=(2t\hat{i}+3t^{2}\hat{j})$ N. The power generated by the force at time $t$ is:

JEE Main · 2025 · 07 Apr 2025 · Shift 1 · Work, Power & Energy

An object of mass 1000 g experiences a time dependent force

F = (2 t \hat{i} + 3 t^{2} \hat{j})

N. The power generated by the force at time

t

is:

(A) $(2 t^{2} + 3 t^{3})$ W
(B) $(2 t^{2} + 18 t^{3})$ W
(C) $(3 t^{3} + 5 t^{5})$ W
(D) $(2 t^{3} + 3 t^{5})$ W✓ Correct

Answer: (D)

Solution

The mass is

m = 1

kg, so the acceleration equals the force in magnitude:

a = \frac{F}{m} = 2 t \hat{i} + 3 t^{2} \hat{j} .

Starting from rest, the velocity is the time integral of the acceleration:

v = \int_{0}^{t} a d t = t^{2} \hat{i} + t^{3} \hat{j} .

Instantaneous power is the dot product of force and velocity:

P = F \cdot v = (2 t) (t^{2}) + (3 t^{2}) (t^{3}) = 2 t^{3} + 3 t^{5} .

The power generated at time

t

(2 t^{3} + 3 t^{5})

JEE Main · 2025 · 07 Apr 2025 · Shift 1 · Single correct option

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