Question

A ball of mass $M$ moving with a speed of $2m/s$ hits another ball of mass $1kg$ moving in the same direction with a speed of $1m/s$. If the kinetic energy of center of mass is $\frac{4}{3}$ Joule, then the magnitude of $M$ is

• A. $1kg$
• B. $0.25kg$
• C. $0.50kg$
• D. $2kg$

Answer 1

Answer: (C)

According to given situation,

As there is no external force, velocity and hence $KE$ of centre of mass remains same before and after collision.
Now,
$v_{CM}=\frac{m_1{v}_1+m_2{v}_2}{m_1+m_2}$
$\Rightarrow v_{CM}=\frac{2M+1}{M+1}$
Kinetic energy of centre of mass is given $\frac{4}{3}J$.
$\therefore \frac{4}{3}=\frac{1}{2}(M+1)\cdot v_{CM}^2$
$\Rightarrow \frac{4}{3}=\frac{1}{2}(M+1){\left(\frac{2M+1}{M+1}\right)}^2$
$\Rightarrow \frac{8}{3}=\frac{(2M+1{)}^2}{M+1}$
$\Rightarrow 8(M+1)=3(2M+1{)}^2$
$\Rightarrow 12M^2+4M-5=0$
$\Rightarrow 12M^2+10M-6M-5=0$
$\Rightarrow 2M(6M+5)-1(6M+5)=0$
$\Rightarrow (2M-1)(6M+5)=0$
$\Rightarrow M=\frac{1}{2}$
or $M=\frac{-5}{6}$
So, mass, $M=\frac{1}{2}=0.5kg$