Q. A particle is projected with certain velocity at two different angels of projections with respect to horizontal plane so as to have same range R on a horizontal plane. If $t_{1}$ and $t_{2}\ $are the time taken for the two paths, the which one of the following relations is correct?
A) $t_{1}t_{2} = \frac{2R}{g}$
B) $t_{1}t_{2} = \frac{R}{g}$
C) $t_{1}t_{2} = \frac{R}{2g}$
D) $t_{1}t_{2} = \frac{4R}{g}$
Explanation
(a) If the horizontal range is the same then the angle of projection of an object is $\theta$ or $(90{^\circ} - \theta)$ with the horizontal direction. So, the angle of projection of first particle is $\theta$ and the other particle is $(90{^\circ} - \theta)$
$t_{1} = \frac{2usin\theta}{g}$
$t_{2} = \frac{2usin\theta(90{^\circ} - \theta)}{g}$
$t_{2} = \frac{2ucos\theta}{g}$
$\because\ \ t_{1}t_{2} = \frac{2usin\theta}{g} \bullet \frac{2ucos\theta}{g}\ $
$t_{1}t_{2} = \frac{2u^{2}\sin 2\theta}{g^{2}}$
$or\ \ \ t_{1}t_{2} = \frac{2R}{g}\ \ \ \ \left( \because R = \frac{u^{2}sin2\theta}{g} \right)\ \ \ \ \ \ \ \ \ \ \ $
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.