An object of mass 1000 g experiences a time dependent force $\vec{F}=\left(2 t \hat{i}+3 t^2 \hat{j}\right)…

An object of mass 1000 g experiences a time dependent force $\vec{F}=\left(2 t \hat{i}+3 t^2 \hat{j}\right) N$. The power generated by the force at time $t$ is :
  1. $\left(2 t^2+3 t^3\right) W$
  2. $\left(2 t^2+18 t^3\right) W$
  3. $\left(3 t^3+5 t^5\right) W$
  4. $\left(2 t^3+3 t^5\right) W$

Solution

$\begin{aligned} & \overrightarrow{\mathrm{F}}=(2 t \hat{\mathrm{i}}+3 \mathrm{t} \hat{\mathrm{j}}) \mathrm{N} \\ & \mathrm{m}=1000 \mathrm{gm}=1 \mathrm{~kg} \\ & \overrightarrow{\mathrm{~F}}=m \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{a}}=2 \mathrm{t} \hat{\mathrm{i}}+3 \mathrm{t}^2 \hat{j} \\ & \frac{\mathrm{~d} \overrightarrow{\mathrm{v}}}{\mathrm{dt}}=2 t \hat{\mathrm{i}}+3 \mathrm{t}^2 \hat{\mathrm{j}}\end{aligned}$
$\vec{v}=t^2 \hat{i}+t^3 \hat{j}$
Power, $\mathrm{P}=\overrightarrow{\mathrm{F}} \cdot \overrightarrow{\mathrm{v}}$
$\begin{aligned} & P=\left(2 t \hat{i}+3 t^2 \hat{j}\right) \cdot\left(t^2 \hat{i}+t^3 \hat{j}\right) \\ & P=\left(2 t^3+3 t^5\right) W\end{aligned}$

Asked in: JEE Main 2025 (07 Apr Shift 1)

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