If two particles have equal momenta, are their kinetic energies equal?

a) yes, always

m1v1 = m2v2 by the question
for KE to equal m1v1^2 = m2v2^2 (I cancelled the 1/2)
But lets call m1v1 = m2v2 = X
so in KE Xv1 = Xv2
so KE are equal if v1 = v2

it’s D. momentum is a product of mass*velocity p = mv.
Object 1 has mass m1 = 10kg and velocity v1 = 10 m/s, so p1 = (10 kg)(10 m/s) = 100 kgm/s.
Object 2 has mass m2 = 5 kg and velocity v2 = 20 m/s, so p2 = (5 kg)(20 m/s) = 100 kgm/s.
Both objects have different masses and velocities, but the same momentum, p1 = p2.
Kinetic energy of Object 1 is KE = 1/2 m1 v1^2 = 1/2 (10 kg) (10 m/s)^2 = 500 J.
Kinetic energy of Object 2 is KE = 1/2 m2 v2^2 = 1/2 (5 kg) (20 m/s)^2 = 1000 J.
Both kinetic energies are different.
This is one example of a proof that proves answer choice D.

Both c and e are correct.
If they have same mass, then they have to have same speed as their momenta is same.
similarly, their mass is same if they have same speed and momenta.
Kinetic energy will same if they have have mass and speed. But only same momenta is not sufficient. Raymond gives a good example below.

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