Kinetic energy of the system of 2 masses....

[Human_being]

Homework Statement

Velocity of centre of mass of two particles is v and the sum of the masses of two particles is m.
Kinetic energy of the system -
(A) will be equal to ½mv²
(B) will always be less than ½mv²
(C) will be greater than or equal to ½mv²
(D) will always be greater than ½mv²

Homework Equations

Kinetic energy = ½mv²

The Attempt at a Solution

I assumed the velocity of particles to be v₁ and v₂ and the mass of particles to be m₁ and m₂ respectively.

Now Kinetic energy of centre of mass = ½mv² = ½(m₁v₁ + m₂v₂)²/ (m₁ + m₂)

And kinetic energy of system = ½( m₁v₁² + m₂v₂²)

Then I tried to compare them and find out which is larger but I couldn't do so.
The only conclusion I came to was that they both will be equal when v₁ = v₂.
How should I proceed now?

 

[lewando]

Try to consider some specific cases.
Case 1: m₁ = m₂, v₁ = -v₂ (m₁ traveling in +x direction, m₂ in the -x direction, same speed).

 

[Human_being]

Try to consider some specific cases.
Case 1: m₁ = m₂, v₁ = -v₂ (m₁ traveling in +x direction, m₂ in the -x direction, same speed).

Then kinetic energy of centre of mass will be zero and that of system will be mv².
So, can I safely conclude from here that kinetic energy of system will always be greater than or equal to kinetic energy of centre of mass?

 

[lewando]

Have you considered any other cases?

 

[haruspex]

can I safely conclude from here that kinetic energy of system will always be greater than or equal to kinetic energy of centre of mass?

You can't assume that based on a single example.

tried to compare them

How? Please post your attempt. And remember that the velocities are vectors.

 

[Human_being]

Have you considered any other cases?

Yes, I considered when either v₁ or v₂ is zero.
Again I got the result that kinetic energy of system is greater than kinetic energy of centre of mass.

 

[Human_being]

How? Please post your attempt. And remember that the velocities are vectors.

As I said I couldn't do it.
What I did was something like this -
½mv² = ½(m₁/m₁ + m₂)m₁v₁² + ½m₂/(m₁ + m₂)m₂v₂² + m₁m₂v₁v₂/m₁ + m₂ - (1)

Kinetic Energy of system = ½m₁v₁² + ½m₂v₂² - (2)

So, first two terms of equation (1) are smaller than the two terms of equation (2) respectively but I couldn't do anything about the third term of equation (1).

 

[haruspex]

As I said I couldn't do it.
What I did was something like this -
½mv² = ½(m₁/m₁ + m₂)m₁v₁² + ½m₂/(m₁ + m₂)m₂v₂² + m₁m₂v₁v₂/m₁ + m₂ - (1)

Kinetic Energy of system = ½m₁v₁² + ½m₂v₂² - (2)

So, first two terms of equation (1) are smaller than the two terms of equation (2) respectively but I couldn't do anything about the third term of equation (1).

So subtract one from the other. To make it easier, multiply both by m first.

 

[Human_being]

So subtract one from the other. To make it easier, multiply both by m first.

Ok, I got it.
When I subtracted (1) from (2) my answer came positive.

Thank you so much haruspex and lewando for helping me. This was my first question and I am happy that it's solved.