# Kinetic energy with respect to different reference frames

You have successfully discovered that the kinetic energy depends on the reference frame.

That is actually true. What is amazing, however, is that the fact that kinetic energy is conserved is NOT reference frame-dependent. So, when you balance your conservation of energy equation in the two frames, you'll find different numbers for the total energy, but you will also see that the energy before and after an elastic collision will be that same number.

So, let's derive the conservation of energy in two reference frames. I'm going to model an elastic collision between two particles. In the first reference frame, I am going to assume that the second particle is stationary, and we have:

\begin{align} \frac{1}{2}m_{1}v_{i}^{2} + \frac{1}{2}m_{2}0^{2} &= \frac{1}{2}m_{1}v_{1}^{2} + \frac{1}{2}m_{2}v_{2}^{2}\\ m_{1}v_{i}^2 &= m_{1}v_{1}^{2} + m_{2}v_{2}^{2} \end{align}

to save myself time and energy, I'm going to call $\frac{m_{2}}{m_{1}} = R$, and we have:

$$v_{i}^{2} = v_{1}^{2} + Rv_{2}^{2}$$

Now, what happens if we shift to a different reference frame, moving to the right with speed v? This is essentially the same thing as subtracting $v$ from all of these terms. We thus have:

\begin{align} (v_{i}-v)^{2} + R(-v)^{2} &= (v_{1}-v)^{2} + R(v_{2}-v)^{2}\\ v_{i}^{2} -2v_{i}v + v^{2} + Rv^{2} &= v_{1}^{2} - 2 vv_{1} + v^{2} + Rv_{2}^{2}-2Rv_{2}v + Rv^{2}\\ v_{i}^{2} -2v_{i}v &= v_{1}^{2}- 2vv_{1} + Rv_{2}^{2}-2Rv_{2}v\\ v_{i}^{2} &= v_{1}^{2} + Rv_{2}^{2} + 2v(v_{i} - v_{1} - R v_{2}) \end{align}

So, what gives? It looks like the first equation, except we have this extra $2v(v_{i} - v_{1} - R v_{2})$ term? Well, remember that momentum has to be conserved too. In our first frame, we have the conservation of momentum equation (remember that the second particle has initial velocity zero:

\begin{align} m_{1}v_{i} + m_{2}(0) &= m_{1}v_{1} + m_{2}v_{2}\\ v_{i} &= v_{1} + Rv_{2}\\ v_{i} - v_{1} - Rv_{2} &=0 \end{align}

And there you go! If momentum is conserved in our first frame, then apparently energy is conserved in all frames!

As you say, energy is not invariant under reference frame change.

Imagine a moving ball. It has kinetic energy, but if I move in its reference frame, it doesn't. It's as simple as this.

There's no need to make up for the missing energy.

In newtonian mechanics, kinetic energy is reference frame dependent.

If this were a relativistic description, the rest mass of the system is invariant under boosts and rotations.