First slide

Projectile motion

Question

Two particles are projected from the same point with the same speed at different angles $θ_{1}$ and $θ_{2}$ to the horizontal. They have the same range. Their times of flight are t₁ and t₂ respectively.

Moderate

A

$\frac{t_{1}}{t_{2}} = \tan^{2} θ_{1}$
B

$\frac{t_{1}}{\sin θ_{1}} = \frac{t_{2}}{\cos θ_{2}}$
C

$\frac{t_{1}}{t_{2}} = \tan θ_{1}$
D

$\frac{t_{1}}{t_{2}} = \tan^{2} θ_{2}$

Solution

$R = \frac{u^{2} \sin 2 θ}{g} is same for angles θ & 90^{\circ} - θ$
$\begin{array}{l} t_{1} = \frac{2 u \sin θ_{1}}{g} = \frac{2 u \sin θ_{1}}{g} \\ t_{2} = \frac{2 u \sin θ_{2}}{g} = 2 u \sin \frac{(90^{\circ} - θ_{1})}{g} \\ \frac{t_{1}}{t_{2}} = \frac{\sin θ_{1}}{\sin (90^{\circ} - θ_{1})} \\ = \frac{\sin θ_{1}}{\cos θ_{1}} = \tan θ_{1} \end{array}$

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