I guess I'm getting old and dumb but I'm having trouble with a basic physics concept, I'm sure you can explain it... Why is kinetic energy 1/2mV^2 and not mV? ***** Example of my confusion no1. Suppose person A has a mass of 1 and a velocity of 10 (on ice skates). Momentum = mV = 10. Kinetic Energy=1/2mV^2 = 50. Person A glides up to person B who also has a mass of 1. Person A hugs person B and they both begin to move as a unit with a total mass of 2 and speed of 5. Momentum is conserved... Momentum = mV = 10. Energy is lost... Kinetic Energy = 1/2mV^2 = 25. Why is energy not conserved? ***** Example of my confusion no2 using the name values... There is a body with mass of 1 floating in space (the body has a rocket engine attached). You are in spaceship A at rest relative to the body in question. I am in spaceship B with a velocity of 10 relative to you and the body. You calculate it's Ek to be 0, I calculate it to be -50. We both watch the object accelerate so that it matches my speed, this required 1lb of fuel. Now you see me and the body traveling at a speed of 10, the body appears at rest to me. I calculate it's Ek to be 0, you calculate it to be 50. Now it accelerates again by the same amount so that it's speed is 20 relative to you, 10 relative to me. From my perspective this again should have taken 1lb of fuel since we changed it's energy by the same amount (-50 to 0 and then 0 to 50).... but to you it took much much more fuel because you saw it's energy increase 4 times (0 to 50 and then 50 to 200). Where is my logic messed up?