# Ball here, Ball there

Classical Mechanics Level pending

A uniform ball of radius $$\text{R}$$ started from rest and descended a vertical distance $$\text{h}$$ while rolling without slipping down a ramp. Then after it hit the ground, the experiment is again conducted but now, ramp is replaced by rails $$d$$ distance apart.

Find the ratio $\displaystyle {(\frac{{v}_{cm, ramp}}{{v}_{cm, rails}})}^{2}$. If it can be written as $$\frac{a}{b}$$, give your answer as $$a + b$$.

Details

1. $$R = 3 m, d = 2 m, h = 10 m, g = 9.8 m{s}^{-2}$$

2. Friction and air resistance are not be considered.

3. $${v}_{cm, a}$$ means velocity of the center of mass when 'a' is used.

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