Question

For a given velocity, a projectile has same range $R$ for two angles of projections. If $t_1$ and $t_2$ are the time of flight in the 2 cases then

• A. $t_1t_2\propto\frac{1}{R^2}$
• B. $t_1t_2\propto\frac{1}{R}$
• C. $t_1t_2\propto R$
• D. $t_1t_2\propto R^2$

Answer 1

Answer: (C)

R = $\frac{u^2 \, sin \, 2\theta_1}{g}$
$ = \frac{u^2 \, sin \, 2\theta_2}{g}$
Also, $\theta_1 + \theta_2 = 90^\circ $
i.e. $\theta_2 = 90^\circ - \theta_1$
Also, $t_1 = \frac{2u\, sin \, \theta_1}{g} $ and
$t_2 = \frac{2u\, sin \, \theta_2}{g}$ then
$t_1 \,t_2 = \frac{4u^2}{g^2} \, sin \, \theta_1 \, sin \, \theta_2$ i.e.
$t_1 \, t_2 = \frac{2}{g} \frac{u^2}{g} 2 \, sin \, \theta_1 \, sin (90 - \theta_1)$
$= \frac{2}{g} \frac{u^2}{g} 2 \, sin \, \theta_1 \, cos \, \theta_1$
$ = \frac{2}{g} \frac{u^2 \, sin \, 2\theta_1}{g}$
$ = \frac{2}{g}R$ i.e. $t_1 \, t_2 \propto R$