Question

Two projectiles each of mass $m$ are projected with same velocity $v$ making an angle $\alpha$ and $\beta$ from the same point in opposite directions. Find the change in their momentum at any instant :

• A. $2mv\sin (\alpha +\beta )$
• B. $2mv\sin \frac{\alpha +\beta }{2}$
• C. $2mv\cos (\alpha +\beta )$
• D. $2mv\cos \frac{\alpha +\beta }{2}$

Answer 1

The correct option is D $2mv\cos \frac{\alpha +\beta }{2}$

Horizontal change in momentum $\Delta p_x=m[v\cos \alpha -(-v\cos \beta )]$

Vertical change in momentum $\Delta p_y=m[v\sin \alpha -gt-(v\sin \beta -gt)]$

Net change in momentum $\Delta p=\sqrt{\Delta p_x^2+\Delta p_y^2}$
$=mv\sqrt{{(\cos \alpha +\cos \beta )}^2+{\left(\sin \alpha -\sin \beta \right)}^2}$

$=mv\sqrt{2+2(\cos \alpha \cos \beta -\sin \alpha \sin \beta )}$
$=2mv\cos \frac{\alpha +\beta }{2}$