A) \[\frac{u}{(g\sin \theta )}\]
B) \[\frac{u}{(gcos\theta )}\]
C) \[\frac{2u}{(g\sin \theta )}\]
D) \[2u\,\tan \theta \]
Correct Answer: A
Solution :
Initial velocity \[\vec{u}=u\cos \theta \,\hat{i}+u\sin \theta \hat{j}\] velocity at time t, \[\vec{v}=u\cos \theta \,\hat{i}+(u\sin \theta -gt)\hat{j}\] \[\therefore \]\[\vec{u}\] and \[\vec{u}\]are perpendicular, \[\vec{u}.\,\vec{v}=0\] \[\Rightarrow \]\[{{u}^{2}}{{\cos }^{2}}\theta +{{u}^{2}}{{\sin }^{2}}\theta -(u\sin \theta )gt=0\] \[\Rightarrow \]\[t=\frac{u}{g\sin \theta }\] Hence, the correction option is (a).
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