### 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 **p**_{1i}.

The momentum of cart 1 after the collision is **p**_{1f} =
**p**_{1i} +Δ**p**_{1}.

The momentum of cart 2 before the collision is **p**_{2i}.

The momentum of cart 2 after the collision is **p**_{2f} =
**p**_{2i} +Δ**p**_{2}.

The total momentum of the system beforehand is **p**_{1i} +
**p**_{2i}.

The total momentum of the system afterwards is **p**_{1f} +
**p**_{2f} = **p**_{1i} + Δ**p**_{1} +
**p**_{2i} +Δ**p**_{2}.

Consider Δ**p**_{1}, 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, Δ**p**_{2}, 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
Δ**p**_{1} = -Δ**p**_{2}.

The total momentum of the system beforehand is **p**_{1i} + **p**_{2i}.

The total momentum of the system afterwards is **p**_{1f} +
**p**_{2f} = **p**_{1i} + Δ**p**_{1} +
**p**_{2i} + Δ**p**_{2} = the total momentum of
the system beforehand!