Work Energy Power

JEE Main level, single correct, medium

144 of 183

Question 144 of 183

An object of mass 1000 g experiences a time dependent force F⃗=(2ti^+3t2j^)N\vec{F}=\left(2 t \hat{i}+3 t^2 \hat{j}\right) N. The power generated by the force at time tt is :

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Answer: (D) (2t3+3t5)W\left(2 t^3+3 t^5\right) W

F→=(2ti^+3tj^)Nm=1000gm=1 kg F→=ma→,a→=2ti^+3t2j^ dv→dt=2ti^+3t2j^\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}
v⃗=t2i^+t3j^\vec{v}=t^2 \hat{i}+t^3 \hat{j}
Power, P=F→⋅v→\mathrm{P}=\overrightarrow{\mathrm{F}} \cdot \overrightarrow{\mathrm{v}}
P=(2ti^+3t2j^)⋅(t2i^+t3j^)P=(2t3+3t5)W\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}

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