Sphere rolling uphill a hollow sphere is rolling along a horizontal floor at 5.0 m/s when it comes to a 30 degree incline. how far up the incline does it roll before reversing direction?

The sphere has translational and rotational kinetic energy.

Trans equation: Ek = ½mv²

Rotation equation: Ek = ½Iω²

substitute I = 2mr²/3

and ω = v / r:

rotat Ek = ½ (2mr²/3) (v/r) ² = (1/3) mv²

Then total E = (½ + 1/3) mv²

= (5/6) mv²

= (5/6) m (5m/s) ²

= m*20.8m²/s²

The sphere attains a vertical height such that this E is converted into potential E:

mgh = m*9.8m/s²*h = m*20.8m²/s²

mass m cancels:

= 20.8 / 9.8

so h = 2.12 m

This translates into a distance along the incline of s = 2.12 m / sin30º = 4.24 m