Probing the misconception:
Two blocks, one with a mass of 1 kg the other with a mass of 2 kg, start from rest. They experience a constant force of 10 N for 1 s. What are their kinetic energies after the force has been applied.
The acceleration of the blocks is given by a = F/m. This gives 10 m/s/s (5 m/s/s) for the smaller (larger) block. Since they accelerate for 1 second their final velocities are 10 m/s (5 m/s). Thus, using 1/2 mv2 their final kinetic energies are 50 J (25 J).
Two blocks, one with twice the mass of the other, start from rest. They experience the same constant force for 1 s. Compare their kinetic energies after the force has been applied.
Since the two blocks experience the same impulse (same force, same time), they must have the same final momenta (via the impulse-momentum theorem). Thus the kinetic energy (p2/2m) of the smaller block must be larger.