Question

From a canon mounted on a wagon at a height $H$ from the ground, a shell is fired horizontally with a velocity $v_0\text{m/s}$ with respect to canon. The canon and wagon have a combined mass of $M\text{kg}$ and can move freely on the frictionless, horizontal surface. Find the horizontal distance (in $\text{metres}$) between shell and canon, when the shell touches ground.

• A. $v_0\sqrt{\frac{2H}{g}}$
• B. $\frac{v_{0}m}{M+m}\sqrt{\frac{2H}{g}}$
• C. $\frac{v_{0}M}{m}\sqrt{\frac{2H}{g}}$
• D. $\frac{v_{0}m}{M}\sqrt{\frac{2H}{g}}$

Answer 1

The correct option is A $v_0\sqrt{\frac{2H}{g}}$

Shell is fired at initial speed $u=v_0\text{m/s}$ in horizontal direction w.r.t the canon.
$\therefore$ solving with respect to canon's frame of reference:-
When shell touches ground, the horizontal displacement of shell

$x_{rel}=u_{rel}t$................(1)
where $t=\text{time of flight for shell}=\sqrt{\frac{2H}{g}}$
$u_{rel}=\text{relative velocity of shell w.r.t}$
$\text{canon in x-direction (Horizontal)}$
Given $u_{rel}=v_0\text{m/s}$

Now sustituting in eq.(1) gives:
$x_{rel}=u_{rel}t$
$\Rightarrow x_{rel}=v_0\times \sqrt{\frac{2H}{g}}=v_0\sqrt{\frac{2H}{g}}\text{metres}$