Two particles A and B are projected simultaneously in a vertical plane as shown in figure. They, collide at time t in air. Write down two necessary equations for collision to take place.

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$(u_{1} \cos θ_{1} + u_{2} \cos θ_{2}) t = 0 a n d (u_{1} \sin θ_{1} - u_{2} \sin θ_{2}) t = 0$

$(u_{1} \cos θ_{1} + u_{2} \cos θ_{2}) t = 20 a n d (u_{1} \sin θ_{1} - u_{2} \sin θ_{2}) t = 10$

$(u_{1} \cos θ_{1} + u_{2} \cos θ_{2}) t = 20 a n d (u_{1} \sin θ_{1} - u_{2} \sin θ_{2}) t = 20$

$(u_{1} \cos θ_{1} + u_{2} \cos θ_{2}) t = 10 a n d (u_{1} \sin θ_{1} - u_{2} \sin θ_{2}) t = 10$

answer is A.

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Detailed Solution

For the two particles to collide at time t, their coordinates should be equal at t.

$∴ x_{A} = x_{B} \Rightarrow 10 + (u_{1} \cos θ_{1}) t = 30 - (u_{2} \cos θ_{2}) t or, (u_{1} \cos θ_{1} + u_{2} \cos θ_{2}) t = 20 Also, y_{A} = y_{B} \Rightarrow 10 + (u_{1} \sin θ_{1}) t - \frac{1}{2} g t^{2} = 20 + (u_{2} \sin θ_{2}) t - \frac{1}{2} g t^{2} or, (u_{1} \sin θ_{1} - u_{2} \sin θ_{2}) t = 10$