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:

s = ut - \frac12 gt^2

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

The horizontal motion of the shell is given by:

s = vt

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

r = vt

At vertical displacement zero, you have:

0 = ut - \frac12 gt^2

Which gives:

0 = u - gt

\frac{u}{g} = t

So, this gives:

r = v(\frac{u}{g})

r = \frac{uv}{g}