Expert: James J. Kovalcin - 11/4/2009

Question
If two particles have equal kinetic energies, are their momenta necessarily equal? Sir Please Explain it.

Answer
Not at all! Kinetic energy is given by: KE=1/2*m*v^2 while momentum is given by: p=m*v
Suppose that one particle has twice the mass of another: m2=2*m1
If their kinetic energies are equal: 1/2*m1*v1^2=1/2*m2*v2^2
Then their velocities will be related according to:
1/2*m1*v1^2=1/2*m2*v2^2
m1*v1^2=m2*v2^2
m1*v1^2=[2*m1]*v2^2
v1^2=2*v2^2
Solve for the velocity of the 2nd particle in terms of the first:
v2=v1/sqrt[2]
Given that the momentum of the first particle is : p1=m1*v1
Then the momentum of the 2nd particle will be: p2=m2*v2=[2*m1]*[v1/sqrt2]=1.41*[m1*v1]
Clearly, m1*v1 is not equal to 1.41*m1*v1!
So although the two particles have equal kinetic energy they do NOT have equal momenta.