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
________.

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

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