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.

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

$$\begin{gathered} \text{∴ }{x}_A=x_B \Rightarrow 10+\left(u_1\cos {\theta }_1\right)t=30-\left(u_2\cos {\theta }_2\right)t \\ \text{or, }\left(u_1\cos {\theta }_1+u_2\cos {\theta }_2\right)t=20 \\ \text{Also, }{y}_A=y_B \Rightarrow 10+\left(u_1\sin {\theta }_1\right)t-\frac{1}{2}g{t}^2=20+\left(u_2\sin {\theta }_2\right)t-\frac{1}{2}g{t}^2 \\ \text{or, }\left(u_1\sin {\theta }_1-u_2\sin {\theta }_2\right)t=10 \end{gathered}$$