Question

For a given velocity, a projectile has the same range $R$ for two angles of projection if $t_1$ and $t_2$ are the time of flight in the two cases, then:

• A. $t_1{t}_2\propto R$
• B. $t_1{t}_2\propto R^2$
• C. $t_1{t}_2\propto \frac{1}{R^2}$
• D. $t_1{t}_2\propto \frac{1}{R}$

Answer 1

Answer: (A)

$t_1{t}_2\propto R$

The range of a projectile motion is given as $R=\frac{2u^{2}sin\theta cos\theta }{g}$
It two projectiles with same velocity have same range,

$R=\frac{2u^{2}sin{\theta }_{1}cos{\theta }_1}{g}=\frac{2u^{2}sin{\theta }_{2}cos{\theta }_2}{g}$

$⟹{\theta }_2={90}^{\circ }-{\theta }_1$

Thus, the times of the flights are:

$t_1=\frac{2usin{\theta }_1}{g}$

$t_2=\frac{2usin{\theta }_2}{g}=\frac{2ucos{\theta }_1}{g}$

Thus, $t_1{t}_2\propto sin{\theta }_{1}cos{\theta }_1\propto R$