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  4. A particle is projected with a velocity u making an angle θ with the horizontal. The instantaneous power of the gravitational force

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Q. A particle is projected with a velocity $u$ making an angle $\theta $ with the horizontal. The instantaneous power of the gravitational force

NTA AbhyasNTA Abhyas 2020Work, Energy and Power

Solution:

At any time, $\overset{ \rightarrow }{\textit{v}}=\left[\left(\textit{u} \text{ cos} \theta \right) \hat{\text{i}} + \left(\textit{u} \text{ sin} \theta - \textit{gt}\right) \hat{\text{j}}\right]$
$\textit{P}=\overset{ \rightarrow }{\textit{F}}\text{.}\overset{ \rightarrow }{\textit{v}}=\left(\text{-} \textit{mg} \hat{\text{j}}\right)\text{.}\left[\textit{u} \text{ cos} \theta \hat{\text{i}} + \left(\textit{u} \text{ sin} \theta - \textit{gt}\right) \hat{\text{j}}\right]$
$=\textit{mg}^{2}\textit{t}-\textit{mgu }\text{sin}\theta $
Hence, power varies linearly with time.