First slide

Projectile motion

Question

A projectile can have same range R for two angles of projection. It t₁ and t₂ are the times of flight in the two cases, then what is the product of two times of flight?

Moderate

A

$t_{1} t_{2} \propto R^{2}$
B

$t_{1} t_{2} \propto R$
C

$t_{1} t_{2} \propto \frac{1}{R}$
D

$t_{1} t_{2} \propto \frac{1}{R^{2}}$

Solution

The horizontal range is the same for the angles of projection $θ and (90^{\circ} - θ)$
$\begin{array}{l} t_{1} = \frac{2 u \sin θ}{g}, t_{2} = \frac{2 u \sin (90^{\circ} - θ)}{g} = \frac{2 u \cos θ}{g} \\ t_{1} t_{2} = \frac{2 u \sin θ}{g} \times \frac{2 u \cos θ}{g} = \frac{2}{g} [\frac{u^{2} \sin 2 θ}{g}] = \frac{2}{g} R \end{array}$
$where R = \frac{u^{2} \sin 2 θ}{g}$
$Hence, t_{1} t_{2} \propto R (as R is constant)$

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