# Conserving energy for a system

Let us assume two bodies which are moving with speeds $$u_1$$ and $$u_2$$. After an elastic collision,we assume the speeds became $$v_1$$ and $$v_2$$ respectively. Now let us treat the objects separately as systems meaning our first object is our system $$1$$ and the second object is system $$2$$. And we know that the energy of a system is always constant by conservation of energy. So why do we write $$\frac{1}{2}m_1u_1^2+\frac{1}{2}m_2u_2^2=\frac{1}{2}m_1v_1^2+\frac{1}{2}m_2v_2^2$$ instead of $$\frac{1}{2}m_1u_1^2=\frac{1}{2}m_1v_1^2$$ and $$\frac{1}{2}m_2u_2^2=\frac{1}{2}m_2v_2^2$$? So,what I am asking is, why can't we use conservation of energy for the two systems separately?

In the case where you chose one body as the system, you ignored the fact that there exists a non conservative external force on the body for a short time during collision.

Conservation of Energy doesn't apply in such cases. What you're doing here is, transferring energy of one system to another which you didn't consider in your equation. Therefore, the error.

You've to have an isolated system for the law to apply.

System 1 - body 1 alone
external force due to body 2
work done by external force due to body 2 equals the change in kinetic energy of body 1

System 2 - body 2 alone
external force due to body 1
work done by external force due to body 1 equals the change in kinetic energy of body 2

System 3 - body 1 and body 2
no external force
no external work done and hence no change in the total kinetic energy of the system

The two individual systems are interacting. Their individual energies are not constant. Only their total energy is constant because the total system is not interacting with anything else.