<Dd> E k = ∫ v 2 2 d m = ∫ v i 2 2 d m + V ⋅ ∫ v i d m + V 2 2 ∫ d m . (\ displaystyle E_ (\ text (k)) = \ int (\ frac (v ^ (2)) (2)) dm = \ int (\ frac (v_ (i) ^ (2)) (2)) dm+ \ mathbf (V) \ cdot \ int \ mathbf (v) _ (i) dm+ (\ frac (V ^ (2)) (2)) \ int dm .) </Dd> <P> However, let ∫ v i 2 2 d m = E i (\ displaystyle \ int (\ frac (v_ (i) ^ (2)) (2)) dm = E_ (i)) the kinetic energy in the center of mass frame, ∫ v i d m (\ displaystyle \ int \ mathbf (v) _ (i) dm) would be simply the total momentum that is by definition zero in the center of mass frame, and let the total mass: ∫ d m = M (\ displaystyle \ int dm = M). Substituting, we get: </P> <Dl> <Dd> E k = E i + M V 2 2 . (\ displaystyle E_ (\ text (k)) = E_ (i) + (\ frac (MV ^ (2)) (2)).) </Dd> </Dl> <Dd> E k = E i + M V 2 2 . (\ displaystyle E_ (\ text (k)) = E_ (i) + (\ frac (MV ^ (2)) (2)).) </Dd>

Where does the kinetic energy formula come from