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A body of mass 1 kg begins to move under the action of a time dependent force $\vec{F} = (2t\hat{i} + 3t^2\hat{j})$ 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)?
A
$(2t^2 + 4t^4)$ W
B
$(2t^3 + 3t^4)$ W
C
$(2t^3 + 4t^5)$ W
D
$(2t + 3t^3)$ W
Explanation
Integrate a(t) to get v(t), then $P = \vec{F}\cdot\vec{v}$.
Detailed Solution
$\vec{a} = \frac{\vec{F}}{m} = 2t\hat{i} + 3t^2\hat{j}$ (m = 1 kg)
$d\vec{v} = (2t\hat{i} + 3t^2\hat{j})dt \Rightarrow \vec{v} = t^2\hat{i} + t^3\hat{j}$
Power $P = \vec{F}\cdot\vec{v} = (2t\hat{i} + 3t^2\hat{j})\cdot(t^2\hat{i} + t^3\hat{j})$
$P = (2t^3 + 3t^5)$ W
Note: the source solution's final line prints $(2t^3 + 4t^5)$ W, which is the keyed option; the dot product actually gives $2t^3 + 3t^5$, which matches none of the options. Please check this question before use.
$d\vec{v} = (2t\hat{i} + 3t^2\hat{j})dt \Rightarrow \vec{v} = t^2\hat{i} + t^3\hat{j}$
Power $P = \vec{F}\cdot\vec{v} = (2t\hat{i} + 3t^2\hat{j})\cdot(t^2\hat{i} + t^3\hat{j})$
$P = (2t^3 + 3t^5)$ W
Note: the source solution's final line prints $(2t^3 + 4t^5)$ W, which is the keyed option; the dot product actually gives $2t^3 + 3t^5$, which matches none of the options. Please check this question before use.