Kinetic energy scales with the square of velocity for the empty rocket.

Steps:

• Initial mass of empty rocket: M; initial speed: v; initial KE = (1/2)Mv².
• Final mass of empty rocket: M; final speed: 2v; final KE = (1/2)M(2v)² = (1/2)M(4v²) = 2Mv².
• Ratio of final to initial KE: [2Mv²] / [(1/2)Mv²] = 4.
• Thus, KE increases by a factor of four.

Why C is correct:

• Kinetic energy formula KE = (1/2)mv² shows doubling velocity multiplies KE by 4 for constant mass.

Why the others are wrong:

• A: Doubling would occur if velocity increased linearly, but KE depends on v².
• B: Halving ignores the velocity increase and quadratic dependence.
• D: KE changes due to velocity doubling, per conservation of momentum in rocket propulsion.

Final answer: C