Power developed by time-dependent force – Rankers Physics

Power: Practice Problem & Solution

A body of mass $1\text{ kg}$ begins to move under the action of a time dependent force $\vec{F} = (2t\hat{i} + 3t^2\hat{j})\text{ N}$, where $\hat{i}$ and $\hat{j}$ are unit vectors along $x$ and $y$ axis. What power will be developed by the force at the time $t$? (2016 - I)
$(2t^2 + 3t^2)\text{W}$
$(2t^2 + 4t)\text{W}$
$(2t^3 + 4t^4)\text{W}$
$(2t^3 + 3t^5)\text{W}$

Solution Explained:

To solve this problem, we apply the core principles of Power. Understanding the underlying formula is key to arriving at the correct answer below:

Velocity is found by integrating acceleration: $\vec{v} = \int \frac{\vec{F}}{m} dt = (t^2\hat{i} + t^3\hat{j})$. Power is given by the dot product $P = \vec{F} \cdot \vec{v} = (2t)(t^2) + (3t^2)(t^3) = 2t^3 + 3t^5\text{ W}$.

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