# A shell is fired vertically upwards with a velocity V1 from the Deck of a ship travelling at a speed of V to a person on the Shore observes the motion of the shell as parabola its horizontal range is given by?

Mar 17, 2018

Actually as the ship is moving horizontally with velocity $v$ so when a shell is fired at velocity ${v}_{1}$ vertically,it also proceeds with horizontal velocity $v$ ,as seen by a person on the shore.

So,its trajectory looks like as follows, Suppose,the height of the ship is $h$,so if the shell took time $t$ to go to the water then,using $s = u t - \frac{1}{2} g {t}^{2}$

we get,

$- h = {v}_{1} t - \frac{1}{2} g {t}^{2}$ (as,if the shell goes up by distance $x$ then on reaching water its total displacement is $x - \left(x + h\right) = - h$)

so,$t = \frac{2 {v}_{1} + \sqrt{4 {v}_{1}^{2} + 8 g h}}{2 g}$ (omited the negative value as, $2 v + k > 2 v$ so that would have given negative value of time)

So,in this time if it had gone distance $R$ horizontally, then,

$R = v t = v \left(\frac{2 {v}_{1} + \sqrt{4 {v}_{1}^{2} + 8 g h}}{2 g}\right)$