Alright. According to relativity, there is no such thing as absolute velocity, only relative velocities. So, we should be able to say the same thing about kinetic energies: Ek=(1/2)mv^2, so since velocity is relative, energy must be as well. But that pesky ^2 presents some difficulties.
Imagine a 1-dimensional universe with 3 objects named m1 m2 and m3, each having a mass of 2kg. Their positions are irrelevant, we're just paying attention to their relative velocities and energies.
So, let's look at a few different possible states for these objects.
For notation, Sn refers to state (S1 is state 1, etc) vab refers to the velocity of ma relative to mb (v12 is velocity of 1 relative to 2) and Eab is the same as vAB except for energy. I won't be mentioning 21, 31 or 32, as those will simply be the negative of 12, 13 and 23 respectively
Starting out, let's say all three masses are stationary with respect to eachother. This means that each one has a velocity of 0m/s relative to both of the other two and a kinetic energy of 0J relative to both of the other two. Remember, Ek=(1/2)mv^2
v12 = v13 = v23 = 0m/s
E12 = E13 = E23 = 0J
Next, let's change the velocity of m1 to 2m/s
v12 = v13 = 2m/s
v23 = 0m/s
E12 = E13 = 4J
E23 = 0J
So far so good.
Next, let's change the velocity of m2 to 1m/s
v12 = 1m/s
v13 = 2m/s
v23 = 1m/s
E12 = 1J
E13 = 4J
E23 = 1J
Now this is where it breaks down. The Energy of 3 relative to 1 should be equal to the sum of the energies of 3 relative to 2 and 2 relative to 1, but it isn't: 1J + 1J != 4J.
Another way to look at it would be to imagine you're in a car which has a total mass of 1000kg with you in it. It starts out stationary relative to the earth. How much energy then does it take to accelerate to 10m/s relative to the earth? If we take the earth as a reference point, then it accelerates from 0m/s to 10m/s, changing its Ek from 0J to 50000J: it takes 50KJ to accelerate the car to 10m/s. What if we take the sun as a reference point, however? The earth itself is travelling at about 30km/s relative to the sun. If you're on the equator and it's the right time of day, you can add almost 0.5km/s to that because of the rotation of the earth, but I'll leave it at 30km/s because that's an easy number to work with. So now, if we're travelling in the same direction as the earth, we're accelerating from 30000m/s to 30010m/s. That's going from 45000000000000J (45TJ) to 450300050000J (45.030005TJ) which is an increase of 30050000J (300MJ.) That is a hell of a lot more than the original 50kJ we calculated, and just a lot of energy period to accelerate a car by 10m/s. But we're not done yet. What happens if we look at the center of the galaxy as our reference point? Our sun travels around the galaxy at a speed of about 235km/s relative to the center. If the earth were at the point in its orbit where it travels in the same direction as the sun, and the car is accelerating in the same direction, that means we're going from 265000m/s to 265010m/s, which translates into an increase in energy of 2650050000J (2.65GJ.) So, how much energy does it take to accelerate the car? 50KJ, 300MJ, 2.65GJ, or something completely different? And why is it that particular one?
The problem intensifies when we consider conservation of energy problems: Any object which is in free fall toward a planet must have a sum of kinetic and gravitational potential energy which remains constant, and for all of these problems, we consider the planet to be stationary even though it clearly is not, and in fact there is no such thing as "stationary."
So, what's going on here? How do we work around this, or fix it?