First slide

Collisions

Question

A particle of mass m is projected with a speed u from the ground at angle $θ = \frac{π}{3}$ w.r.t. horizontal ( x-axis). When it has reached its maximum height, it collides completely inelastically with another particle of the same mass and velocity $u \hat{i}$ . The horizontal distance covered by the combined mass before reaching the ground is

Moderate

A

$\frac{3 \sqrt{3}}{8} \frac{u^{2}}{g}$
B

$\frac{5}{8} \frac{u^{2}}{g}$
C

$\frac{3 \sqrt{2}}{4} \frac{u^{2}}{g}$
D

$2 \sqrt{2} \frac{u^{2}}{g}$

Solution

According to Conservation of Momentum,
$\frac{m u}{2} + m u = 2 m v^{1} \Rightarrow v^{1} = \frac{3 u}{4}$
Range of combined mass $R^{1} = v^{1} \sqrt{\frac{2 H}{g}}$ $= \frac{3 u}{4} \sqrt{\frac{2. H}{g}}$ $= \frac{3 u}{4} . \sqrt{\frac{2. u^{2} \sin^{2} 60}{g .2 g}}$ $= \frac{3 u}{4} . \sqrt{\frac{2. u^{2} .3}{4. g .2 g}}$ = $\frac{3 \sqrt{3}}{8} \frac{u^{2}}{g}$

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