Question

conceptual example #3: a rock is being dropped from the top of a cliff that is 30 meters high as shown in the pictures on the right. at what point does the rock have both kinetic and potential energy, but more kinetic energy than potential energy? (a) point a (b) point b (c) point c (d) point d (e) point e

Explanation:

Step1: Recall Energy Concepts

Potential energy (PE) depends on height (\(PE = mgh\)), kinetic energy (KE) on speed (\(KE=\frac{1}{2}mv^2\)). Total mechanical energy (\(E = PE + KE\)) is conserved (no air resistance). At the top (Point A), \(KE = 0\), \(PE\) is maximum. As the rock falls, \(PE\) decreases, \(KE\) increases.

Step2: Analyze Height vs Energy

• At \(h = 30m\) (Point A): \(PE_{max}\), \(KE = 0\) (no motion yet).
• At \(h = 15m\) (Point C): \(PE = \frac{1}{2}mgh_{total}\) (since \(h = 15 = \frac{30}{2}\)), \(KE=\frac{1}{2}mgh_{total}\) (because \(E = mgh_{total}\), so \(KE = E - PE = mgh_{total}-\frac{1}{2}mgh_{total}=\frac{1}{2}mgh_{total}\)). Here, \(PE = KE\).
• Below \(h = 15m\) (e.g., Point D, E), \(h < 15m\), so \(PE = mgh < \frac{1}{2}mgh_{total}\), and \(KE = E - PE > \frac{1}{2}mgh_{total}\). Thus, \(KE > PE\) at points below 15m (like Point D, E). But among the options, Point D is below 15m (since Point C is at 15m). Wait, let's check the diagram: Point A (top, 30m), B (above C), C (15m), D (below C), E (ground). So at Point D, height is less than 15m, so \(PE < KE\) (since \(KE = E - PE\), and \(E = 30mg\) (at A, \(E = PE = 30mg\)). At D, \(PE = mgh_D\), \(h_D < 15\), so \(PE < 15mg\), \(KE = 30mg - mgh_D > 15mg\), so \(KE > PE\). At Point C, \(PE = KE\) (both \(15mg\)). At Points B (above C), \(h > 15\), so \(PE > KE\) (since \(KE = 30mg - mgh_B\), \(h_B > 15\) implies \(mgh_B > 15mg\), so \(KE < 15mg\), \(PE > 15mg\)). At Point A, \(KE = 0\), \(PE = 30mg\). So the point with \(KE > PE\) is below 15m, i.e., Point D (or E, but E is ground, \(PE = 0\), \(KE = 30mg\), but the options have D and E. Wait, the options are A, B, C, D, E. So Point D is below C (15m), so at D, \(KE > PE\).

Answer:

D) Point D