Question

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

• A. $337.5\text{ W}$
• B. $228.5\text{ W}$
• C. $422.5\text{ W}$
• D. $126.5\text{ W}$

Hint:

If force depends on time, first find velocity by integrating acceleration, then use: \[ P=\vec{F}\cdot\vec{v} \]

Answer 1

The Correct Option is A

Concept:
Instantaneous power is: \[ P=\vec{F}\cdot \vec{v} \] Since mass is \(1\text{ kg}\), \[ \vec{a}=\vec{F} \] ip

Step 1: Find velocity components by integrating acceleration.
Given: \[ \vec{a}=t\hat{i}+2t^2\hat{j} \] Since the body begins to move from rest: \[ v_x=\int t\,dt=\frac{t^2}{2} \] \[ v_y=\int 2t^2\,dt=\frac{2t^3}{3} \] So, \[ \vec{v}=\frac{t^2}{2}\hat{i}+\frac{2t^3}{3}\hat{j} \] ip

Step 2: Write force and velocity at \(t=3\text{ s}\).
At \(t=3\): \[ \vec{F}=3\hat{i}+18\hat{j} \] \[ \vec{v}=\frac{9}{2}\hat{i}+\frac{2\cdot 27}{3}\hat{j} =\frac{9}{2}\hat{i}+18\hat{j} \] ip

Step 3: Find power.
\[ P=\vec{F}\cdot\vec{v} \] \[ P=3\left(\frac{9}{2}\right)+18(18) \] \[ P=\frac{27}{2}+324 \] \[ P=13.5+324=337.5\text{ W} \] ip Hence, the correct answer is:
\[ \boxed{(A)\ 337.5\text{ W}} \]