Question

A particle of mass $2kg$ moving with a velocity $5\hat{i}m/s$ collides head-on with another particle of mass $3kg$ moving with a velocity $-2\hat{i}m/s$. After the collision the first particle has speed of $1.6m/s$ in negative $x$ direction. Find the :
1)velocity of the centre of mass after the collision
2)velocity of the second particle after the collision
3)coefficient of restitution.

Answer 1

Given that,

Mass $m_1=2kg$

Mass $m_2=3kg$

Initial velocity $u_1=5\hat{i}m/s$

Initial velocity $u_2=-2\hat{i}m/s$

Final velocity $v_1=-1.6m/s$

Now, by conservation of momentum

$m_1{u}_1+m_2{u}_2=m_1{v}_1+m_2{v}_2$

$2\times 5\hat{i}+3\times -2\hat{i}=2\times -1.6\hat{i}+3v_2$

$v_2=2.4\hat{i}m/s$

Velocity of center of mass after collision

$=\frac{m_1\times v_1+m_2\times v_2}{m_1+m_2}$

$=\frac{2\times -1.6\hat{i}+3\times 2.4\hat{i}}{5}$

$=\frac{4\hat{i}}{5}$

Coefficient of restitution is

$v_1-v_2=-e\times (u_1-u_2)$

$-e=\frac{v_1-v_2}{u_1-u_2}$

$e=\frac{1.6+2.4}{5+2}$

$e=\frac{4}{7}$

$e=0.57$

Hence, the velocity after collision is $\frac{4\hat{i}}{5}$ and coefficient of restitution is $0.57$