Two objects have unequal masses, m1>m2; which object has greater momentum?

[DavidAp]

Two objects have unequal masses, m1 > m2. If their kinetic energies are equal, which has the greater momentum?

This is the way I approached this problem. I know that,
Momentum = mass*velocity
K = 1/2mv^2

So, I solved for v since it is the only unknown in these two equations.
k = 1/2mv^2
2k/m = v^2
√2k/m = v

And plugged it into the first equation for momentum.
Momentum = m√2k/m
= √m^2*√2k/m
= √m^2 2k/m
= √2k m

However, since 2k is a constant between the two massess I removed them from the equation giving me Momentum = √m

Since m1>m2 I thought it was safe to say that the momentum of object 1 is greater than the momentum of object 2; however, I was wrong. According to the answer the momentum of m2 is greater than m1 so I'm just wonder, why?

Is my math wrong, am I not suppose to join these two equation? If somebody can help I would greatly appreciate it. Thank you for taking the time to review my question.

 

[Delphi51]

Fascinating problem and solution!
You have momentum = constant * sqrt(m).
So the one with the bigger m has the bigger momentum.
Makes sense.

Another approach is the say ½m1v1² = ½m2v2²
so m1*v1 = m2*v2²/v1
momentum1 = m2*v2*v2/v1
momentum 1 = momentum2*v2/v1
The first line ½m1v1² = ½m2v2² with m1 > m2 implies v1 < v2
so v2/v1 is larger than 1 and momentum1 must be larger than momentum2.

Your conclusion has been found correct by two methods . . .
A single counterexample can disprove the textbook answer.
Say m1=4, v1 = 1, m2=1, . Then v2 = 2 to make the energies equal.
p1 = 4*1 = 4, p2 = 1*2 = 2.