Question

part a rank speed from greatest to least at each point. rank from greatest to least. to rank items as equivalent, overlap them. buttons: d, c, e, b, a; ranking boxes labeled greatest to least; checkbox: the correct ranking cannot be determined; buttons: submit, request answer; image of roller coaster with points a, b, c, d, e

Explanation:

Step1: Analyze Potential Energy and Kinetic Energy

In a roller - coaster motion (assuming no non - conservative forces like friction for simplicity, or at least using the principle of conservation of mechanical energy), the total mechanical energy \(E = K+U\) (where \(K\) is kinetic energy and \(U\) is potential energy) is conserved. Kinetic energy \(K=\frac{1}{2}mv^{2}\), and potential energy (gravitational) \(U = mgh\) (where \(h\) is the height above a reference point). So, the lower the height \(h\), the lower the potential energy \(U\), and the higher the kinetic energy \(K\), and thus the higher the speed \(v\) (since \(v=\sqrt{\frac{2K}{m}}\) and mass \(m\) is constant for the roller - coaster car).

Looking at the roller - coaster diagram:

• Point D is at the lowest height among the points A, B, C, D, E. So, the potential energy at D is the lowest, and the kinetic energy (and thus speed) is the highest.
• Point B: After point A (the starting point, which is at a high height), the car moves down to B. The height of B is lower than A, C, and E, but higher than D. So, the speed at B is less than at D but more than at points with higher heights.
• Point E: The height of E is higher than B and D, so the potential energy at E is higher than at B and D, so the kinetic energy (and speed) at E is less than at B and D.
• Point C: The height of C is higher than B and D (and similar to or higher than E? From the diagram, C is a peak, E is also a peak but maybe C is at a lower height than E? Wait, looking at the diagram, D is the lowest trough, B is a trough (but higher than D), C is a peak (higher than B but lower than A and E?), E is a peak (higher than C), and A is the starting peak (highest). Wait, maybe I misread. Let's re - evaluate:
• The height order (from highest to lowest) is: \(h_A>h_E > h_C>h_B>h_D\)
• Since \(v\propto\sqrt{\frac{2(E - U)}{m}}\) and \(E\) is constant (conservation of energy, ignoring friction), the lower the height \(h\) (lower \(U\)), the higher the speed \(v\). So the speed order (from greatest to least) is based on the inverse of the height order. So speed order: \(v_D>v_B>v_E>v_C>v_A\)

Step2: Rank the Speeds

Based on the analysis of the heights (and thus the relationship between potential and kinetic energy), the speed from greatest to least is D (greatest), then B, then E, then C, then A (least).

Answer:

From greatest to least: D, B, E, C, A