First slide

Projectile motion

Question

For a given velocity, a projectile has the same range R for two angles of projection if t₁ and t₂ arc the times of flight in the two cases then

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

For same range angle of projection should be $θ$ and, 90- $θ$
$So, time of flights t_{1} = \frac{2 u \sin θ}{g} and$
$t_{2} = \frac{2 u \sin (90 - θ)}{g} = \frac{2 u \cos θ}{g}$
$By multiplying t_{1} t_{2} = \frac{4 u^{2} \sin θ \cos θ}{g^{2}}$
$t_{1} t_{2} = \frac{2}{g} \frac{(u^{2} \sin 2 θ)}{g} = \frac{2 R}{g} \Rightarrow t_{1} t_{2} \propto R$

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