Which of the following frictionless ramps (a, b, or c) will give the ball the greatest speed at the bottom of the ramp? explain.

Let the ball is initially at point P1. Finally it reaches at point P2.( bottom most). Since only conservative force is acting on the ball throughout the motion, mechanical energy will be conserved. Therefore conserving energy from P1 to P2, Initially ball is at rest(P1). So initial kinetic energy is 0. Let ball is at height h1 initially, so initial potential energy = mgh ( m= mass of the ball). At bottom most point (P2), potential energy of ball changes into kinetic energy. So final kinetic energy =1/2 mv^2. Final potential energy = 0. Equating energies at P1 and P2, Mgh= 1/2 mv^2. Therefore, v= (2gh)^1/2. Since all the initial and final points are at same height, velocity will be same for all the paths(irrespective of the path taken by ball). So velocity will be same.

It should be d if not then c