Question

A particle of mass $m$ moving in the $x$-direction with speed $2v$ is hit by another particle of mass $2m$ moving in the $y$-direction with speed $v$. If the collision is perfectly inelastic, the percentage loss in the energy during the collision is close to

• A. 50 %
• B. 56 %
• C. 62 %
• D. 44 %

Answer 1

Answer: (B)

before collision
${\vec{P}}_x=2mv\hat{i}$
${\vec{P}}_y=2mv\hat{j}$

After collision
${\vec{P}}_x=3m{v}^{\prime }\cos \theta$
${\vec{P}}_y=3v^{\prime }\sin \theta$
By momentum conservation ;
in horizontal $\to 2mv=3mv\cos \theta \dots (i)$
in vertical $\to 2mv=3m{v}^{\prime }\sin \theta \dots (ii)$
from (i) and (ii) $\tan \theta =1;\theta =45^{\circ }$
final speed $v^{\prime }=\frac{2\sqrt{2}v}{3}$
$\text{ initial }K.E.;\to 1/2(m)(2v{)}^2+1/2(2m)(v{)}^2=3m{v}^{\prime 2}$
final $K.E.;\to 1/2(3m){\left(\frac{2\sqrt{2}v}{3}\right)}^2=4/3m{v}^2$
$\%$ loss $\to \frac{(KE{)}_1-(KE{)}_f}{(KE{)}_i}\times 100\%$
$=55.55\simeq 56\%$