If the momentum of a given particle is doubled, then its kinetic energy will be
The momentum of a body is given as the product of its mass and velocity. Mathematically,
p = mv
where, p is momentum, m is the mass of the body and v is its velocity.
The kinetic energy of a body is directly proportional to the mass of the body and square of its velocity. Mathematically,
K.E. = `1/2 mv^2`
In this case, momentum is doubled. Assuming that the mass of the particle remains constant, only a change in velocity can bring about a change in the particle's momentum. So, momentum can only be doubled if the velocity doubles.
If, p' is the new momentum and
p' = 2 p = 2 mv = mv'
means that the new velocity is twice the old velocity.
And the new kinetic energy can be calculated as
K.E.' = `1/2 m v'^2 = 1/2 m (2v)^2 = 4 [1/2mv^2] = 4 K.E.`
i.e., the new kinetic energy is 4 times the older kinetic energy.
Hope this helps.