Kinetic energy scales with the square of velocity for constant mass

Steps:

• Initial mass of empty rocket: M; initial speed: v; initial KE = (1/2) M v².
• After firing, empty rocket mass remains M; final speed: 2v; final KE = (1/2) M (2v)² = (1/2) M (4v²) = 2 M v².
• Factor of increase: final KE / initial KE = [2 M v²] / [(1/2) M v²] = 4.
• Fuel expulsion provides thrust but does not alter the rocket body's mass for KE calculation.

Why C is correct:

• Kinetic energy formula KE = (1/2) m v² shows doubling velocity quadruples KE for fixed mass M.

Why the others are wrong:

• A: Doubling would require linear velocity scaling, but KE is quadratic.
• B: Halving ignores velocity increase and quadratic dependence.
• D: Speed change from v to 2v cannot keep KE constant per KE formula.

Final answer: C