First slide

Projection Under uniform Acceleration

Question

It is possible to project a particle with a given velocity in two possible ways so as to make it pass through a point at a distance r from the point of projection. The product of times taken to reach this point in the two possible ways is then proportional to

Easy

A

r
B

r²
C

1/r
D

1/r²

Solution

The range of projectile
$r = \frac{u^{2} \sin 2 θ}{g} = \frac{2 u^{2} \sin θcos θ}{g}$
The two possible angles of projection are $θ and (90^{\circ} - θ) .$
The times of flight corresponding to these angles are
$t_{1} = \frac{2 usin θ}{g}$
and $t_{2} = \frac{2 usin (90^{\circ} - θ)}{g} = \frac{2 ucos θ}{g}$
$\begin{array}{l} ∴ t_{1} t_{2} = \frac{4 u^{2} \sin θcos θ}{g^{2}} = \frac{2 r}{g} \\ or t_{1} t_{2} \propto r \end{array}$

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