Question

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

Moderate

Solution

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

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