Kinetic energy depends on mass and velocity squared

Steps:

• Kinetic energy formula: KE = ½ m v².
• Let mass of Y be m, speed of Y be v; then mass of X is 2m, speed of X is v/2.
• KE_Y = ½ m v².
• KE_X = ½ (2m) (v/2)² = ½ (2m) (v²/4) = (m v²)/4 = ½ (½ m v²), so KE_X = ½ KE_Y.

Why A is correct:

• Ratio KE_X/KE_Y = (m_X/m_Y) × (v_X/v_Y)² = 2 × (½)² = 2 × ¼ = ½, matching the formula KE = ½ m v².

Why the others are wrong:

• B: Ignores mass doubling, which offsets half the speed-squared reduction (¼ factor).
• C: Overemphasizes mass doubling, ignoring speed halving that squares to ¼ effect.
• D: Fails to account for both mass increase and speed decrease impacting KE differently.

Final answer: A