# Q & A: Masses, Momentum, and Kinetic Energy

Q:
if a large mass and small mass have the same momentum, which has the greater kinetic energy?
- Murk
England
A:
Let's have 2 objects, X1 and X2. They have the masses m1 and m2, with X1 being the big mass and X2 being the small mass, so X1 is the larger object. Their momentums are p1 and p2, respectively. To keep vectors out of this problem, let's assume that both masses are traveling parallel to each other in the same direction.

Keep in mind that momentum equls mass (in kg) times velocity (in m/s).

If p1 = p2, and m1 >> m2, then that means that v1 << v2, by the same magnitude.

For the momentums to be equal, the product of the velocities and masses of the 2 objects must be equal, so the velocity of X2 must counter the larger mass of X1

So if the small mass is 1 kg and the large mass is 6 kg, and the large mass travels at 2 m/s, then the small mass must be going at 12 m/s to make the momenta equal.

The non-relativistic formula for Kinetic energy is KE = 1/2 mv^2, with m being the mass in kg and v being the velocity, in m/s.

Because the velcity term is squared, any fractional change to the velocity will also be squared when computing the kinetic energy. So if the velocity is doubled, the kinetic energy is quadrupled.

Using our previous example, the KE for X1 would be:

= 1/2 (6 kg) (2 m/s)^2 = 12 J (kg m^2/s^2)

For X2, that would be:

= 1/2 (1 kg) (12 m/s)^2 = 72 J (kg m^2/s^2)

So, with the conditions that I have described, the smaller mass would have more kinetic energy than the larger mass, at that instant.

Jason

A short way to see this is that if p=mv is the momentum, and KE=(1/2)mv^2 is the nonrelativistic kinetic energy, then KE=p^2/(2m) is another way of writing the kinetic energy. If p is the same for the two objects, KE is bigger for the one with the smaller m (smaller denominator). The inequality is the same for the relativistic case too, but the expressions are a little more complicated.

Tom

(published on 10/22/2007)