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Definition of a Projectile

Two particles...

Question

Two particles are projected from a point at the same instant with velocities whose horizontal and vertical components are $u_{1}, v_{1}$ and $u_{2}, v_{2}$ respectively. The interval between their passing through the other common point of their path is

A

\frac{2 (v_{1} u_{1} - v_{2} u_{2})}{g (u_{1} + u_{2})}

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B

\frac{2 (v_{1} u_{1} + v_{2} u_{2})}{g (u_{1} + u_{2})}

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C

\frac{2 (v_{2} u_{1} - v_{1} u_{2})}{g (u_{1} + u_{2})}

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D

\frac{2 (v_{1} v_{2} - u_{1} u_{2})}{g (u_{1} + u_{2})}

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Solution

The correct option is C $\frac{2 (v_{2} u_{1} - v_{1} u_{2})}{g (u_{1} + u_{2})}$
$x = u_{1} t = u_{2} (t + Δ t) - \frac{1}{2} g (t + Δ t)^{2}$ ,
$y = v_{1} t - \frac{1}{2} g t^{2} = v_{2} (t + Δ t) - \frac{1}{2} g (t + Δ t)^{2}$
$\Rightarrow Δ t = \frac{2 (v_{2} u_{1} - v_{1} u_{2})}{g (u_{1} + u_{2})}$

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