Question

A body of mass m$_{1}$ ​=4 kg moves at 5i m/s and another body of mass m$_{2}$ =2 kg moves at 10i m/s. The kinetic energy of centre of mass is
$\begin{align}
  & A.\dfrac{200}{3}J \\
 & B.\dfrac{500}{3}J \\
 & C.\dfrac{400}{3}J \\
 & D.\dfrac{800}{3}J \\
\end{align}$

Answer 1

Hint: Kinetic energy is the product of mass and velocity square multiplied with half, now we have the mass of each body individually and the velocity of each body separately, we have to calculate the velocity at the center of mass then find the value of Kinetic energy.
Velocity of center of mass can be found out,
\[{{V}_{CM}}=\dfrac{{{m}_{1}}{{v}_{1}}+{{m}_{2}}{{v}_{2}}}{{{m}_{1}}+{{m}_{2}}}\],
 $m{}_{1}$=mass of the first body,
${{v}_{1}}$= velocity of the first body,
${{v}_{2}}$= velocity of the second body,
$m{}_{2}$= mass of the second body.


Complete step-by-step answer:
According to the question we have to calculate the kinetic energy of the center of mass.
So we know that kinetic energy for center of mass is,
\[K.{{E}_{CM}}=\dfrac{1}{2}{{M}_{CM}}\times {{V}_{CM}}^{2}\], here\[{{M}_{CM}}\], is the mass of the bodies, and \[{{V}_{CM}}\] is the velocity of the bodies.
Now we know that ,
\[{{V}_{CM}}=\dfrac{{{m}_{1}}{{v}_{1}}+{{m}_{2}}{{v}_{2}}}{{{m}_{1}}+{{m}_{2}}}\],
 $m{}_{1}$=mass of the first body,
${{v}_{1}}$= velocity of the first body,
${{v}_{2}}$= velocity of the second body,
$m{}_{2}$= mass of the second body.
On putting the values,
\[{{V}_{CM}}=\dfrac{4\times 5\hat{i}+2\times 10\hat{i}}{4+2}\],
\[{{V}_{CM}}=\dfrac{40}{6}\hat{i}\],
\[{{V}_{CM}}=\dfrac{20}{3}\hat{i}\].
Therefore the kinetic energy is:
\[K.{{E}_{CM}}=\dfrac{1}{2}\times \left( 6 \right)\times {{\left( \dfrac{20}{3} \right)}^{2}}J\],
\[K.{{E}_{CM}}=\dfrac{400}{3}J\].
Therefore option c is the correct option.

Additional Information:
Kinetic energy of a body is defined as the energy possessed by the body by virtue of motion.
The unique point where the weighted relative position of the distributed mass sums to zero which is for a distribution of mass in space is known as the centre of mass.
The velocity of an object is the rate of change of position with respect to frame of reference which is a function of time.

Note: \[\hat{i}\] is the unit vector in the direction of i it helps us to determine the direction of the vector. Here we need the velocity and mass at the point of center of mass hence we have to calculate it.