The sport of skateboarding provides an excellent example of the principle of Conservation of Energy. In particular, let us consider 'vert skateboarding' where a person rides the skateboard on a vertical ramp that forms part of a hemisphere referred to as a 'half-pipe.' It consists of the transition from the curved part to the flat and the vertical. Below is a schematic of a half-pipe with the 'vert'. The surface of the half-pipe and the material of the wheels on the skateboard allow for an almost frictionless ride. Therefore we will neglect friction in the following analysis.

The rider starts from rest at location at the edge of the in-ramp and goes down the transition. Typically, as the rider approaches the flat at location he will crouch down to get his center of mass as low as possible and thus increase his speed. To simplify the problem, let us initially assume that the rider stays upright as he goes down so that his center of mass location relative to his feet does not change from what it was at location . In the following problems, you can use the foot as the unit length instead of meters. Note: g = 32 ft/s2.

(c) For the rest of this problem consider just the motion of the center of mass of the rider. Consider a standard half-pipe with h = 11 ft with a 2 ft vert. Consider a 5.5 ft person whose center of mass is at his midheight. He starts from rest at location . What is his speed v2 at location if he does not crouch (his center of mass is at the same height relative to his feet)?

(d) Based on your answer in part (c) above, how high (what is the location of his center of mass relative to location ) will he go up the vert on the other side?

(e) If he wishes to go up beyond the edge of the vert, he will need to have greater kinetic energy when he is at location . To do this, the rider will pump as he goes back and forth on the half-pipe several times (he will crouch down as he approaches the flat to lower his center of mass) to further decrease his potential energy then spring back up to convert his internal muscular energy to kinetic energy. Suppose the rider in this case wishes to rise above the edge of the vert so that the change in his center of mass position is 19 ft relative to the flat section of the pipe (location ), what speed does he need to have when he is at location ?