Question

A body of mass 1 kg begins to move under the action of a time dependent force $\overrightarrow{F}=(2t\hat{i}+3t^2\hat{j})$ N, where $\hat{i}$ and $\hat{j}$ are unit vectors along x and y axes. What power will be developed by the force at the time t?

• A. $(2t^2+3t^2)W$
• B. $(2t^4+4t^4)W$
• C. $(2t^3+3t^4)W$
• D. $(2t^3+3t^5)W$

Answer 1

Answer: (D)

$(2t^3+3t^5)W$

As given $\overrightarrow{F}=(2t\hat{i}+3t^2\hat{j})N$

Where $\hat{i}$ and $\hat{j}$ are unit vectors along x and y axis

$F=ma$

$\Rightarrow a=\frac{F}{m}=\frac{(2t\hat{i}+3t^2\hat{j})}{2}=[m=1kg]\Rightarrow a=(2t\hat{i}+3t^2\hat{j})m/s^2$

Acceleration $a=\frac{dv}{dt}$

$\Rightarrow dv=a.dt\to \left(i\right)\int dv=\int a.dt=\int (2t\hat{i}+3t^2\hat{j})dtv=t^2\hat{i}+t^3\hat{j}P=F.v=(2t\hat{i}+3t^2\hat{j})(t^2\hat{i}+t^3\hat{j})=2t.t^{+}3t^2{t}^{3}P=(2t^3+3t^5)W$