Derive the relation between momentum and kinetic energy.

Explanation

(i) Consider an object of mass m moving with a velocity `vec(v)`. Then its linear momentum is `vec(p)=mvec(v)` and its kinetic energy, `KE1/2mv^(2)`.
`KE=1/2mv^(2)=1/2m(vec(v).vec(v))" ...(1)"`
(ii) Multiplying both the numerator and denominator of equation (1) by mass, m
`KE=1/2(m^(2)(vec(v).vec(v)))/m`
`=1/2((mvec(v)).(mvec(v)))/m[vec(p)=mvec(v)]`
`=1/2(vec(p)*vec(p))/m`
`=p^(2)/(2m), KE=p^(2)/(2m)`
(iii) where `abs(vec(p))` is the magnitude of the momentum. The magnitude of the linear momentum can be obtained by
`abs(vec(p))=p=sqrt(2m(KE))`
(iv) Note that if kinetic energy and mass are given, only the magnitude of the momentum can be calculated but not the direction of momentum. It is because the kinetic energy and mass are scalars.