question_answer

A body is projected at such angle that the horizontal range is three times the greatest height. The angle of projection is

A)  \[{{42}^{o}}8\]                

B)  \[{{53}^{o}}7\]

C)  \[{{33}^{o}}7\]

D)         \[{{25}^{o}}8\]

Correct Answer: B

Solution :

Let a body be projected at a velocity u at an angle \[\theta \] with the horizontal. Then horizontal range covered is given by \[R=\frac{{{u}^{2}}\sin 2\theta }{g}\]                      ?(i) and height H is \[H=\frac{{{u}^{2}}{{\sin }^{2}}\theta }{2g}\]       ?(ii) Given,                   \[R=3H\] \[\therefore \]  \[\frac{{{u}^{2}}\sin 2\theta }{g}=3\times \frac{{{u}^{2}}{{\sin }^{2}}\theta }{2g}\] Also,      \[\sin 2\theta =2\sin \theta \cos \theta \] \[\therefore \]  \[\frac{{{u}^{2}}2\sin \theta \cos \theta }{g}=3\times \frac{{{u}^{2}}{{\sin }^{2}}\theta }{2g}\] \[\Rightarrow \]               \[2\cos \theta =1.5\sin \theta \] \[\Rightarrow \]               \[\tan \theta =\frac{2}{1.5}=1.33\] \[\Rightarrow \]               \[\theta ={{53}^{o}}7\] Hence, angle of projection is \[{{53}^{o}}7\]