A shell is fired vertically upwards with a velocity u from the deck of a ship travelling at a speed of v. A person on the shore observes the motion of the shell as a parabola whose horizontal range is given by.. :)

Well, you can easily derive it.

The initial vertical velocity is u, the initial horizontal velocity is v.

The motion of the shell in the vertical plane is given by:



(I put minus because g is in the opposite direction as to u)

The horizontal motion of the shell is given by:



But to avoid confusion, we'll use r instead of s, because the horizontal displacement is also the range of the shell.



At vertical displacement zero, you have:



Which gives:





So, this gives: