A stationary but hot object has kinetic energy due the the motion of the individual atoms that make it up, even though its overall momentum is 0. I.e.

∑ⱼ mⱼ v⃗ⱼ = 0⃗

where the mⱼ are the masses of the parts of the object and the v⃗ⱼ are the velocities of those parts.

If the object initially has 0 velocity, its kinetic energy is:

T = ½∑ⱼ mⱼ v⃗ⱼ²

Now we give the object a kick (or just switch reference frames) to change its velocity by Δv⃗. The new kinetic energy is:

T' = ½∑ⱼ mⱼ (v⃗ⱼ + Δv⃗)²

T' = ½∑ⱼ mⱼ (v⃗ⱼ² + 2v⃗ⱼ⋅Δv⃗ + Δv⃗²)

T' = ½(∑ⱼ mⱼ v⃗ⱼ²) + Δv⃗⋅(∑ⱼ mⱼ v⃗ⱼ) + ½Δv⃗²(∑ⱼ mⱼ)

If M is the total mass of the object, then we can substitute this into the sum in the last term. And we already saw that the sum in the middle term was 0. So:

T' = ½(∑ⱼ mⱼ v⃗ⱼ²) + Δv⃗⋅0⃗ + ½Δv⃗² M

T' = ½∑ⱼ mⱼ v⃗ⱼ² + ½MΔv⃗²

So in terms of the original kinetic energy T, which was purely thermal energy, we get:

T' = T + ½MΔv⃗²

In other words, because of the quadratic kinetic energy formula, we can see that the total kinetic energy T' of a hot object is just its thermal kinetic energy T plus the usual mechanical kinetic energy ½MΔv⃗².