Final answer:
To find the roller coaster's speed at half its initial height, we use the conservation of energy principle where the sum of potential and kinetic energy remains constant during a frictionless descent.
Step-by-step explanation:
The question is about finding the speed of a roller coaster at a certain point during its descent from the highest point to ground level. By applying the principles of conservation of energy, we know that the mechanical energy (potential + kinetic) at the top must equal the mechanical energy at any other point in the frictionless descent. We can use the equations for potential energy (PE = mgh) and kinetic energy (KE = 0.5mv²) to solve this problem.
At the halfway point, the potential energy is halved (since height h is halved), but the total mechanical energy must remain the same. Thus, the kinetic energy must have increased by the amount that the potential energy decreased. We can solve this set of equations to find the speed of the roller coaster when it is at half the height of its starting point.