# If the momentum of a given particle is doubled, then its kinetic energy will be

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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.