A body of mass 2 kg begins to move under the influence of time dependent force , where and are unit vectors along and -axis respectively

Question

A body of mass 2 kg begins to move under the influence of time dependent force $\vec{F} = (2 t \hat{i} + 6 t^{2} \hat{j}) N$ , where $\hat{i}$ and $\hat{j}$ are unit vectors along $x$ and $y$ -axis respectively. The power produced by the force at $t = 2 s$ is _______ $W$ . [2026]

Smart Achievers · Accepted Answer

(200)

$Given: mass of body = 2 kg$

$\vec{F} = (2 t \hat{i} + 6 t^{2} \hat{j}) N$

$acceleration \vec{a} = \frac{\vec{F}}{m} = \frac{2 t \hat{i} + 6 t^{2} \hat{j}}{2} = (t \hat{i} + 3 t^{2} \hat{j}) {ms}^{- 2}$

$\vec{v} = \int_{0}^{2} \vec{a} \cdot d t = {[\frac{t^{2}}{2} \hat{i} + \frac{3 t^{3}}{3} \hat{j}]}_{0}^{2} = (2 \hat{i} + 8 \hat{j}) m / s$

${\vec{F}}_{t = 2 s} = (4 \hat{i} + 24 \hat{j}) N$

$P = \vec{F} \cdot \vec{v} = 8 + 192 = 200 W$

Solution

Q.
A body of mass 2 kg begins to move under the influence of time dependent force $\vec{F} = (2 t \hat{i} + 6 t^{2} \hat{j}) N$ , where $\hat{i}$ and $\hat{j}$ are unit vectors along $x$ and $y$ -axis respectively. The power produced by the force at $t = 2 s$ is _______ $W$ . [2026]

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Solution

Q. A body of mass 2 kg begins to move under the influence of time dependent force F→=(2ti^+6t2j^) N, where i^ and j^ are unit vectors along x and y-axis respectively. The power produced by the force at t=2 s is _______ W. [2026]

Ans. (200) Given: mass of body=2 kg F→=(2ti^+6t2j^) N acceleration a→=F→m=2ti^+6t2j^2=(ti^+3t2j^) ms-2 v→=∫02a→·dt=[t22i^+3t33j^]02=(2i^+8j^) m/s F→t=2s=(4i^+24j^) N P=F→·v→=8+192=200 W

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Q.
A body of mass 2 kg begins to move under the influence of time dependent force $\vec{F} = (2 t \hat{i} + 6 t^{2} \hat{j}) N$ , where $\hat{i}$ and $\hat{j}$ are unit vectors along $x$ and $y$ -axis respectively. The power produced by the force at $t = 2 s$ is _______ $W$ . [2026]