Let,

**Initial Momentum** =* P*_{i} = m_{i}v_{i} and,

**Initilal Kinetic Energy **= *KE* _{ i }= 1/2 m_{i}v_{i} ^{2}

Now, it is given that the momentum is doubled.

Thus, **New Momentum** = *P*_{f} = m_{f}v_{f} = 2P_{i} = 2m_{i}v_{i}

From here, ** ***v*_{f} = 2m_{i}v_{i}/m_{f}

And, **New Kinetic Energy** = *KE*_{f} = 1/2 m_{f}v_{f} ^{2} = 1/2 m_{f} x (2m_{i}v_{i}/m_{f})^{2} = 1/2 x 4 x m_{f}/m_{f} ^{2} x m _{ i } ^{ 2 } vi^{2}

= 2 x m^{2}/m^{2} x mv _{ i } ^{ 2 } = 2 mv _{ i } ^{ 2 } = 4 x KE _{ i }

Note that *m*_{i} = m_{f} = m as mass doesn 't change in this case.

Thus, the kinetic energy becomes 4 times the initial.

(EDITED)