### Colliding Carts

Two carts, labeled cart 1 and cart 2, collide with one another on a horizontal, essentially frictionless track. How does the momentum of a cart change? What happens to the momentum of the system consisting of the two carts?

Let's use a subscript i for initial and f for final.

The momentum of cart 1 before the collision is p1i.
The momentum of cart 1 after the collision is p1f = p1ip1.

The momentum of cart 2 before the collision is p2i.
The momentum of cart 2 after the collision is p2f = p2ip2.

The total momentum of the system beforehand is p1i + p2i.

The total momentum of the system afterwards is p1f + p2f = p1i + Δp1 + p2ip2.

Consider Δp1, the change in momentum experienced by cart 1 in the collision. This comes from the force applied on cart 1 by cart 2 during the collision (it's the area under the force vs. time graph).

Similarly, Δp2, the change in momentum experienced by cart 2 in the collision, comes from the force applied on cart 2 by cart 1 during the collision (it's the area under that force vs. time graph).

How do the areas under the force vs. time graphs compare?

The areas have equal magnitudes and opposite signs, so Δp1 = -Δp2.

The total momentum of the system beforehand is p1i + p2i.

The total momentum of the system afterwards is p1f + p2f = p1i + Δp1 + p2i + Δp2 = the total momentum of the system beforehand!