A body of mass 1 kg begins to move under the action of a time dependent force $\vec{F}=\left(t \hat{i}+3 t^{2} \hat{j}\right)$ N, where $\hat{i}$ and $\hat{j}$ are the unit vectors along $x$ and $y$ axis. The power developed by above force, at the time $t=2$ s, will be ______ W.

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Work, Power & Energy

A body of mass 1 kg begins to move under the action of a time dependent force

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

N, where

\hat{i}

and

\hat{j}

are the unit vectors along

x

and

y

axis. The power developed by above force, at the time

t = 2

s, will be ______ W.

Answer: 100

Solution

With

m = 1

kg the acceleration equals the force, so the velocity is

v = \int_{0}^{t} (t \hat{i} + 3 t^{2} \hat{j}) d t = \frac{t ^{2}}{2} \hat{i} + t^{3} \hat{j}

. The power delivered is

P = F \cdot v = (t \hat{i} + 3 t^{2} \hat{j}) \cdot (\frac{t ^{2}}{2} \hat{i} + t^{3} \hat{j}) = \frac{t ^{3}}{2} + 3 t^{5}

. At

t = 2

s this is

\frac{8}{2} + 3 (32) = 4 + 96 = 100

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Integer answer

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