Velocity of centre of mass of two particles is v and the sum of the masses of two particles is m The total kinetic energy of the system:
(A) will be equal to 1/2 mv^2
(B) will always be less than 1/2 mv^2
(C) will be greater than or equal to 1/2 mv^2
(D) will always be greater than 1/2 mv^2

The answer given in my book is (C)
I tried using the approach:
Total K.E. of system= K.E(centre of mass) + K.E(of masses with respect to the centre of mass)
but I'm not really able to understand the situation here.
Could someone possibly help me understand this by giving a few examples?
 A: Your intuition is correct. The kinetic energy with respect to the center of mass is what causes the total kinetic energy to be bigger than $\tfrac 1 2 mv^2$. As an example consider two particles connected with a string. Let's assume they have the same mass for a second so $m_1=m_2=\tfrac 1 2 m$. In general these particles could be rotating around the center of mass. Let's call the velocity at which they are rotating with respect to the center of mass $u$. Since their masses are the same their velocity with respect to the center of mass is $u$ for both particles (but in opposite direction). At some point during the rotation $u$ is parallel to $v$. When this happens the velocity of the top particle is $v+u$ and of the bottom one it's $v-u$.

We can calculate the total kinetic energy now.
\begin{align}
KE&=\tfrac 1 2 m_1(v+u)^2+\tfrac 1 2 m_2(v-u)^2\\
&=\tfrac 1 2 m_1 v^2+\tfrac 1 2 m_1u^2+m_1 v u\\
&\ +\tfrac 1 2 m_2 v^2+\tfrac 1 2 m_2u^2-m_1 v u\\
&=\tfrac 1 2(m_1+m_2)v^2+\tfrac 1 2 m_1u^2+\tfrac 1 2 m_2u^2+uv(\underbrace{m_1-m_2}_{0})\\
&=\tfrac 1 2mv^2+\tfrac 1 2 m_1u^2+\tfrac 1 2 m_2u^2
\end{align}
So this (long) calculation was just a way to really see what's going on. The short answer would be that $\text{KE}_{\text{system}}=\text{KE}_{\text{COM}}+\text{KE}_{\text{relative}}$. The kinetic energy of the center of mass is $\tfrac 1 2mv^2$ and since kinetic energy is always positive we can see that the total kinetic energy is always bigger than or equal to $\tfrac 1 2mv^2$.
A: Examples: a football thrown with a spin, the earth-moon system, a car with rotating wheels.
