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All correct. To add, if you double the momentum, which part are you doubling? If you double the mass, you would double the momentum and double the kinetic energy. If you double the velocity, you double the momentum and quadriple (4x) the kinetic energy. If you double both the mass and velocity, you would increase the momentum 4 times and increase the kinetic energy 8 times.
The momentum of a body is given as the product of its mass and velocity at which it is traveling, whereas the kinetic energy is given as one-half of the product of its mass and the square of the velocity with which it is traveling.
Thus, momentum P = m x v
and K.E. = 1/2 m x v^2
from the equation of momentum, v = P/m and substituting this value of velocity in the equation for kinetic energy we get:
K.E. = 1/2 m x v^2 = 1/2 m x (p/m)^2 = `p^2 /(2m)`
Therefore the kinetic energy varies as the square of the momentum. So, if a particle is able to retain its mass, kinetic energy will vary as the square of momentum. For example, a doubling of momentum will increase the kinetic energy by a factor of 4 (it will be 4-times the original value, the increase would be 3-times).
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