question_answer

Two projectiles are projected with the same velocity. If one is projected at an angle of \[30{}^\circ \] and the other at \[60{}^\circ \] to the horizontal. The ratio of maximum heights reached is

A) \[1:3\]               

B) \[2:1\]

C) \[3:1\]               

D) \[1:4\]

Correct Answer: A

Solution :

If body is projected with initial velocity u at an angle \[\theta \] with the horizontal, then the height H reached by projectile is given by    \[H=\frac{{{u}^{2}}\,{{\sin }^{2}}\,\theta \,}{2\,g}\] where g is acceleration due to gravity.          Given, \[{{\theta }_{1}} = 30{}^\circ  and {{\theta }_{2}} = 60{}^\circ \] \[\therefore \,\,\,\,\,\,\frac{{{H}_{1}}}{{{H}_{2}}}=\frac{{{\sin }^{2}}{{\theta }_{1}}}{{{\sin }^{2}}{{\theta }_{2}}}=\frac{{{\sin }^{2}}\,(30{}^\circ )}{{{\sin }^{2}}\,(60{}^\circ )}=\frac{{{\left( \frac{1}{2} \right)}^{2}}}{{{\left( \frac{\sqrt{3}}{2} \right)}^{2}}}=\frac{1}{3}\]