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 \(a + b\).


  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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