Understanding Kinetic Energy and Momentum Conservation in Colliding Masses

[Yosty22]

In my theoretical mechanics class, we went over some very basic conservation laws (namely momentum) and talked about colliding masses.

Say you have 2 masses, m₁ and m₂. m₁ is moving to the right (towards m₂) with velocity v. m₂ is stationary. After the two masses collide, m₁ recoils with velocity -v (same speed, just the direction changed). In order for momentum to be conserved, mass m₂ must move to the right after the collision with speed v'. In terms of v, m₁, and m₂, v' can be described as:
v' = 2m₁v / m₂.

This is all well and good, but some classmates and I were thinking about the kinetic energy here. If m₁ is moving to begin with, it has some initial kinetic energy equal to .5m₁v². However, since it recoils at the same exact speed (just different direction), it's kinetic energy is the same after the collision as it was before. However, m₂ must move to the right after the collision for momentum to be conserved, so after the collision, m₂ also has kinetic energy equal to .5m₂v². That is, the kinetic energy after the collision is greater than the kinetic energy before. (I know kinetic energy doesn't always have to be conserved, but why would the total kinetic energy be greater after this collision?)

 

[Orodruin]

Your observation is correct. If colliding with a stationary object, a mass with initial velocity v can only have final velocity -v if the collision is elastic *and* the object it collides with has infinite mass (or rather, the velocity will go to -v as the mass ratio goes to 0). In this limit, the heavy mass can absorb the additional momentum without changing its kinetic energy significantly.

If the objects have the same mass and the collision is elastic, the originally moving object will be at rest after the collision.