Finding the time elapsed between the instant when two particles are projected and the instant they collide So say you have a vertical mast of height $32$m, and then say a particle P is projected horizontally from the top of the mast at $18$ m/s. Then say that a particle Q is projected at $30$ m/s from the bottom of the mast at an angle x above the horizontal. Given that the particles collide and that $\cos(x) = \frac35$. Is there a way to find the time taken for the collision from the moment they were projected?

The laws of motion are \begin{align} x_1(t)&=v_1^0t\\\ y_1(t)&=h-\frac{1}{2}gt^2\\\ x_2(t)&=v_2^0\cos(x)t\\\ y_2(t)&=v_2^0\sin(x)t-\frac{1}{2}gt^2\\\ \end{align} We can see that $x_1(t) = x_2(t)$ is an identity.
Equating $$ y_1(t)=y_2(t) $$ we get $$ t=\frac{h}{v_2^0\sin(x)}=\frac{32m}{30m/s\sqrt{1-\left(\frac{3}{5}\right)^2}}=1.33s $$ A graphic representation

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