Question 299: Centre of Mass and Rotational Motion

Question

A particle of mass m is moving along the x-axis
with initial velocity $u\widehat i$. It collides elastically
with a particle of mass 10 m at rest and then
moves with half its initial kinetic energy (see
figure). If $\sin {	heta _1} = \sqrt n \sin {	heta _2}$ then value of n is
________.
![](https://imagex.cdn.examgoal.net/jgy2xukexyysik/a2a46eba-b336-4e3b-9c06-bb805a84b88f/76fb71c0-f407-11ea-8293-3b6cc591631c/file-jgy2xukexyysil-393w.png)

Accepted Answer

${1 \over 2}mv_1^2 = {1 \over 2}\left( {{1 \over 2}m{u^2}} ight)$
$ \Rightarrow $ $v_1^2 = {{{u^2}} \over 2}$
$ \Rightarrow $ ${v_1} = {u \over {\sqrt 2 }}$  ......(1)
Also collision is elastic : k
i
= k
f
${{1 \over 2}m{u^2} = {1 \over 2}mv_1^2 + {1 \over 2}\left( {10m} ight)v_2^2}$
$ \Rightarrow $ ${{1 \over 2} 	imes {1 \over 2}m{u^2} = {1 \over 2}\left( {10m} ight)v_2^2}$
$ \Rightarrow $ ${v_2^2 = {{{u^2}} \over {20}}}$
$ \Rightarrow $ ${v_2} = {u \over {\sqrt {20} }}$
By momentum conservation along y :
m
1
v
1
sin $	heta $
1
= m
2
v
2
sin $	heta $
2
$ \Rightarrow $ mv
1
sin $	heta $
1
= 10mv
2
sin $	heta $
2
$ \Rightarrow $ ${u \over {\sqrt 2 }}$sin $	heta $
1
= 10 $ 	imes $ ${u \over {\sqrt {20} }}$sin $	heta $
2
$ \Rightarrow $ sin $	heta $
1
= $\sqrt {10} $ sin $	heta $
2
$ 	herefore $ n = 10
![](https://imagex.cdn.examgoal.net/1ki92tzvl/928bbc1a-2749-4fe7-9a05-78ee61d54c16/db6b9510-3587-11eb-8c16-afaa84271520/file-1ki92tzvm-320w.png)

Centre of Mass and Rotational Motion

A particle of mass m is moving along the x-axis with initial velocity u\widehat i. It collides elastically with a particle of mass 10 m at rest and then moves with half its initial kinetic energy (see figure). If \sin {\theta _1} = \sqrt n \sin {\theta _2} then value of n is ________.

Integer / Numeric Answer Type