1. The problem statement, all variables and given/known data A frictionless track is to be built as shown, with L=4.60m and H=4.80m. In order to get the cart to slide from the top to the end of the track in the minimum time, how long should the distance D be? Assume that the speed of the cart on the horizontal surface is the same as at the bottom of the ramp. 2. Relevant equations 3. The attempt at a solution No friction means energy is conserved. Average velocity is distance divided by time. using conservation laws, i get the following equation .5mv1^2 + mgy1 = .5mv2^2 + mgy2 i set up my coordinates so that y1 is the top of the ramp, and y2 is the end, so y2=0 and we're left with .5mv1^2 + mgy1 = .5mv2^2. the problem wants the D to minimize time, so somehow i need to express the above equation in terms of time, take a derivative, and solve setting=0. i know v(av)=d/t, so can i substitute that expression into .5mv1^2 + mgy1 = .5mv2^2 and get .5m(d/t)^2 + mgy1 = .5m(d/t)^2 since the velocity is the same at the bottom and top of the ramp?