Average Kinetic Energy of 2 Objects

Question

If I had 2 objects of equal mass ( 10kg for example ), and knew that the average speed of the 2 objects was 8 ms^-1 , is it possible to calculate the average kinetic energy of the 2 objects. I attempted the speed of object 1 to be 6 ms^-1 and the speed of Object 2 to be 10 ms^-1 . So the average kinetic energy would be (180 + 500)/2 = 340 Joules , however that is not equal to 320 when you use the average speed so 1/2 * 10 * 8². I understand about root mean squared, but if you don't know the initial values of the speed but only the average is it just not possible to calculate average kinetic energy?

Answer 1

I think you showed a perfect example of why we cannot solve for this without the initial values. Let's prove it!

Let K₁ = (1/2)mv₁², and K₂ = (1/2)mv₂² for the respective objects that share the same mass m, and let K = (1/2)mv², where v = (v₁ + v₂)/2 is the average velocity and K is the kinetic energy of the average velocity. We want to prove that the kinetic energy of the average velocity is not equal to the average kinetic energy. Or otherwise stated: K ≠ (K₁ + K₂)/2

K = (1/2)mv² = (1/2)m[(v₁ + v₂)/2]² = (1/2)m(v₁² + 2v₁v₂ + v₂²)/4

(K₁ + K₂)/2 = [(1/2)mv₁² + (1/2)mv₂²]/2 = (1/2)m(v₁² + v₂²)/2

So see how these two values are not the same. We do know the value of (v₁² + 2v₁v₂ + v₂²)/4, it's 64. But we're left with two variables in one equation, where you could solve for one velocity and plug it into (v₁² + v₂²)/2 in terms of the other.

Answer 2

The short answer is that terms like "average velocity" or "average energy" typically have no real purpose.

With energy, we're typically concerned with totals or sums and differences.

If you have 3 masses travelling at 3 different velocities, the total energy is m1(v1)² + m2(v2)² + m3(v3)².

Looking at that, I can't see how to get total energy--or even average energy--by using average velocity.