Two particles A and B are projected from ground towards each other with speeds 10 m/s and $5\sqrt{2} \mathrm{m}/\mathrm{s}$ at angles $30°$ and $45°$ with horizontal from two points separated by a distance of 15 m. Will they collide or not?

For collision to take place, relative velocity of A with respect to B should be along the line joining them, i.e., their vertical components should be same.

Vertical component of velocity of of A,  ${\left(u_A\right)}_y=10\sin 30°=5 \mathrm{m}/\mathrm{s}$

and vertical component of velocity of B,  ${\left(u_B\right)}_y=5\sqrt{2}\sin 45°=5 \mathrm{m}/\mathrm{s}$.

Since ${\left(u_A\right)}_y={\left(u_B\right)}_y$, the particles may collide.

The second condition for collision of A and B is the sum of their horizontal ranges should be more than or equal to the distance (d = 15 m) between them, i.e., 

$$\begin{gathered} R_A+R_B\ge d\text{.} \\ {R}_A=\frac{(10{)}^2\sin 60°}{10}=8.66 \mathrm{m} \\ {R}_B=\frac{(5\sqrt{2}{)}^2\sin 90°}{10}=5 \mathrm{m} \\ \therefore {\mathrm{R}}_{\mathrm{A}}+{\mathrm{R}}_{\mathrm{B}}=13.66 \mathrm{m} \end{gathered}$$

Since $R_A+R_B<d$, the particles they will not collide.