Question

A body of mass m moving with a velocity v is approaching a second mass but at rest . The kinetic energy of the two objects as viewed from the center is ----

Answer 1

Velocity of centre of mass:

$$\begin{gathered} v_{cm}=\frac{m_1{v}_1+m_2{v}_2}{m_1+m_2} \\ =\frac{mv+m\left(0\right)}{m+m} \\ =\frac{v}{2} \end{gathered}$$

Kinetic energy as viewed from the centre:

$$\begin{gathered} KE=\frac{1}{2}m{\left(v_{cm}\right)}^2 \\ =\frac{1}{2}m{\left(\frac{v}{2}\right)}^2 \\ =\frac{m{v}^2}{8} \end{gathered}$$