A marble of mass M and radius R rolls without slipping down a track from height h1. The marble then goes up a frictionless track to a height h2. Find h2.
I figure this has to be in terms of other variables, here is my work.
initial mechanical energy = final mechanical energy
Mgh1 = Mgh2 + (1/2)Mv^2 + (1/2)Iω^2
I for sphere = (2/5)MR^2
Plug in for I and cancel M’s. Plug in v = ωR
gh1 = gh2 + (1/2)v^2 + (1/5)v^2
g(h1 – h2) = (7/10)v^2
h2 = h1 – (7/10)(v^2 / g)
Can the answer be expressed simpler?