To answer this question we must decide how the kinetic energy is being changed. Kinetic energy depends on both the mass and the speed of the object in question. We can change the kinetic energy by changing either the one or the other, or both. To simplify the answer we will assume we will change either the mass or the speed, but not both.
To double the kinetic energy by changing the mass we look at the the equation for KE and see that it is directly proportional to the mass:
`KE = 1/2mV^2` ; examination of the momentum equation shows that it too is directly proportional to the mass: `p = mV`
therefore, doubling the kinetic energy by doubling the mass will also double the momentum.
If we double the kinetic energy by changing the speed, we notice that because KE is proportional to `V^2`
to double the KE we must increase the velocty by a factor of `sqrt2`
Because momentum is directly proportional to the speed as well as mass, doubling the kinetic energy by changing the speed will increase the momentum by a factor `sqrt2` , or make it `sqrt2` times larger.