K.E=p^2/2m.when p(momentum ) is doubled.,then kinetic energy increases by 4 times....as k E , is directly proportional to p^2.

Kinetic Energy and Momentum Relationship
In order to understand how the kinetic energy of a body changes when its momentum is doubled, we need to first establish the relationship between kinetic energy and momentum.

Kinetic Energy (KE) Formula:
The kinetic energy of an object can be calculated using the formula: KE = 1/2 * m * v^2, where m is the mass of the object and v is its velocity.

Momentum (p) Formula:
Momentum is defined as the product of an object's mass and velocity, given by the formula: p = m * v.

How Momentum is Related to Kinetic Energy:
Since momentum is directly proportional to velocity, doubling the momentum would mean doubling the velocity of the object while keeping the mass constant. This implies that the kinetic energy of the object would increase by a factor of 4 (2^2) when its momentum is doubled.

Derivation:
Let's assume the initial momentum of the body is p and the initial kinetic energy is KE. When the momentum is doubled to 2p, the new kinetic energy will be KE' = 1/2 * m * (2v)^2 = 4 * KE.

Conclusion:
Therefore, when the momentum of a body is doubled, its kinetic energy increases by a factor of 4. This relationship highlights the strong connection between momentum and kinetic energy in the dynamics of moving objects.