Final answer:
The minimum velocity needed at the top of the 16 m tall hill for a roller-coaster to reach the top of a 20 m hill can be found using conservation of energy.
Step-by-step explanation:
The minimum velocity that the roller-coaster would need when going over the 16 m hill to make it to the top of the 20 m hill can be determined using the principle of conservation of mechanical energy, assuming negligible air resistance and friction. We consider the potential energy at the top of the 16 m hill to be equal to the potential energy at the top of the 20 m hill since we want the roller-coaster to just barely reach the top of the 20 m hill without any excess speed.
At the top of the 16 m hill, the roller-coaster's total mechanical energy will be the sum of its potential energy (PE) due to height and kinetic energy (KE) due to velocity, given by PE16m + KE = m*g*h16m + 0.5*m*v16m^2. At the top of the 20 m hill, the mechanical energy will be purely potential, since we are looking for the scenario where the velocity is zero, given by PE20m = m*g*h20m.
Setting these two energies equal to each other and cancelling out the mass (m) as it appears on both sides, we get g*h16m + 0.5*v16m^2 = g*h20m. Solving for v16m yields v16m = sqrt(2*g*(h20m - h16m)). Plugging in g = 9.81 m/s2, h20m = 20 m, and h16m = 16 m gives us the minimum starting velocity at the 16 m hill.