Question

this diagram represents a roller coaster. the cart is pulled to the top of the first hill at point q and then released. which statement correctly describes the carts kinetic energy and gravitational potential energy? a) at point s, gravitational potential energy and kinetic energy are equal. b) at point r, both kinetic energy and gravitational potential energy are increasing. c) at point s, kinetic energy and gravitational potential energy both begin to decrease. d) at point r, kinetic energy is increasing and gravitational potential energy is decreasing.

Explanation:

• Gravitational potential energy (\(U = mgh\), where \(m\) is mass, \(g\) is acceleration due to gravity, and \(h\) is height) depends on the height of the cart. Kinetic energy (\(K=\frac{1}{2}mv^{2}\), where \(v\) is velocity) depends on the speed of the cart.
• When the cart moves down (like at point \(R\) in the roller - coaster motion), its height \(h\) decreases (so gravitational potential energy \(U\) decreases) and its speed \(v\) increases (so kinetic energy \(K\) increases) due to the conversion of gravitational potential energy into kinetic energy (assuming no non - conservative forces like friction are considered in a basic analysis).
• At point \(S\), as the cart moves up, its height \(h\) increases (gravitational potential energy \(U\) increases) and its speed \(v\) decreases (kinetic energy \(K\) decreases).
• For option A, there is no information to suggest that \(U = K\) at point \(S\).
• For option B, at point \(R\) (moving down), \(U\) decreases and \(K\) increases, not both increase.
• For option C, at point \(S\) (moving up), \(U\) increases and \(K\) decreases, not both decrease.

Answer:

D. At point R, kinetic energy is increasing and gravitational potential energy is decreasing.