Two particles are simultaneously projected in the same vertical plane from the same point with velocities u and v at angles $\alpha$ and $\beta$ with horizontal. Find the time that elapses when their velocities are parallel.

$$\begin{gathered} v_1=\left(u\cos \alpha \right)\hat{i}+(u\sin \alpha -gt)\hat{j} \\ {v}_2=\left(v\cos \beta \right)\hat{i}+(v\sin \beta -gt)\hat{j} \end{gathered}$$

These two velocity vectors will be parallel when the ratio of coefficients of $\hat{\mathbf{i}}$ and $\hat{\mathbf{j}}$ are equal.

$\therefore \frac{u\cos \alpha }{v\cos \beta }=\frac{u\sin \alpha -gt}{v\sin \beta -gt}$

Solving, we get

$t=\frac{uv\sin (\alpha -\beta )}{g(v\cos \beta -u\cos \alpha )}$