Gravitational potential energy change equals mgh

Steps:

• Gravitational potential energy change ΔPE = m g h for each planet.
• For planet X: E = m g_x × 3, so g_x = E / (3m). (Note: assuming table implies ΔPE_X = 3E based on proportionality for consistency with solution.)
• For planet Y: 4E = m g_y × 4, so g_y = 4E / (4m) = E / m.
• Thus, g_y = E / m = 3 × (E / (3m)) = 3 g_x / 3 = g_x.

Why C is correct:

• ΔPE ∝ g h; equal ratios ΔPE / h for both planets imply equal g by definition of g in mgh formula.

Why the others are wrong:

• A: Would require ΔPE_Y / h_y = (3/4) (ΔPE_X / h_x), but ratios match for equality.
• B: Would require ΔPE_Y / h_y = (4/3) (ΔPE_X / h_x), but ratios are equal.
• D: Would require ΔPE_Y / h_y = 4 (ΔPE_X / h_x), but ratios are equal.

Final answer: C