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
$\vec{ P }_{ x }=2\, mv \,\hat{ i }$
$\vec{ P }_{ y }=2 \,mv \,\hat{ j }$

After collision
$\vec{ P }_{ x }=3 \,mv' \cos \theta$
$\vec{ P }_{ y }=3\, v ' \sin \theta$
By momentum conservation ;
in horizontal $\rightarrow 2 \,m v=3\, m v \cos \,\theta \,\,\,\,\,\,\ldots(i)$
in vertical $\rightarrow 2 \,mv =3 \,mv ' \sin \,\theta \,\,\,\,\,\ldots(ii)$
from (i) and (ii) $\tan \theta=1 ; \theta=45^{\circ}$
final speed $v'=\frac{2 \sqrt{2} \,v}{3}$
$\text { initial } K . E . ; \rightarrow 1 / 2(m)(2 v)^{2}+1 / 2(2 m)(v)^{2}=3 m v^{\prime 2}$
final $K.E. ; \rightarrow 1 / 2(3 m )\left(\frac{2 \sqrt{2} v }{3}\right)^{2}=4 / 3 mv ^{2}$
$\%$ loss $\rightarrow \frac{( KE )_{1}-( KE )_{ f }}{( KE )_{i}} \times 100 \%$
$=55.55 \simeq 56 \%$