Rotating ball flies certain height in the air

[nhmllr]

Homework Statement

(answer is h * 5 /7)

Homework Equations

gravitational potential energy = mgh
kinetic energy = 1/2 mv²
rotational kinetic energy of sphere = 2/5 mv²

The Attempt at a Solution

The gravitation potential energy at the beginning at the top is the same as the kinetic energy and rotational energy at the bottom

mgh_i = 1/2 mv² + 2/5 mv² = 9/10 mv²

We also know that the rotational kinetic energy at the apex of it's final height, plus the potential energy it achieves there, is equal to the initial potential energy

mgh_i = mgh + 2/5 mv²
h = h_i - (2/5 v²)/g

and from our first equation, we know
mgh_i = 9/10 mv²
4/9 gh_i = 2/5 v²

So combining those...
h = h_i - (4/9 gh_i)/g
h = 5/9 h_i

But that is not the answer... So where did I go wrong?
Thanks


EDIT: The rotational kinetic energy is actually 1/5 mv². Sorry, it makes sense

 

[jbsteele]

now.The Attempt at a SolutionThe gravitational potential energy at the beginning at the top is the same as the kinetic energy and rotational energy at the bottommghi = 1/2 mv2 + 1/5 mv2 = 7/10 mv2We also know that the rotational kinetic energy at the apex of it's final height, plus the potential energy it achieves there, is equal to the initial potential energymghi = mgh + 1/5 mv2h = hi - (1/5 v2)/gand from our first equation, we know mghi = 7/10 mv23/7 ghi = 1/5 v2So combining those...h = hi - (3/7 ghi)/gh = 5/7 hi