The two cars together form the system that is to be analyzed. It is important to remember that the contents (the mass) of the system do not change before, during, or after the objects in the system interact.
Problem-solving strategy: conservation of momentum
Using conservation of momentum requires four basic steps. The first step is crucial:
Identify a closed system (total mass is constant, no net external force acts on the system).
Write down an expression representing the total momentum of the system before the “event” (explosion or collision).
Write down an expression representing the total momentum of the system after the “event.”
Set these two expressions equal to each other, and solve this equation for the desired quantity.
Colliding carts
Two carts in a physics lab roll on a level track, with negligible friction. These carts have small magnets at their ends, so that when they collide, they stick together (
[link] ). The first cart has a mass of 675 grams and is rolling at 0.75 m/s to the right; the second has a mass of 500 grams and is rolling at 1.33 m/s, also to the right. After the collision, what is the velocity of the two joined carts?
Two lab carts collide and stick together after the collision.
Strategy
We have a collision. We’re given masses and initial velocities; we’re asked for the final velocity. This all suggests using conservation of momentum as a method of solution. However, we can only use it if we have a closed system. So we need to be sure that the system we choose has no net external force on it, and that its mass is not changed by the collision.
Defining the system to be the two carts meets the requirements for a closed system: The combined mass of the two carts certainly doesn’t change, and while the carts definitely exert forces on each other, those forces are internal to the system, so they do not change the momentum of the system as a whole. In the vertical direction, the weights of the carts are canceled by the normal forces on the carts from the track.
Solution
Conservation of momentum is
Define the direction of their initial velocity vectors to be the +
x -direction. The initial momentum is then
The final momentum of the now-linked carts is
Equating:
Substituting the given numbers:
Significance
The principles that apply here to two laboratory carts apply identically to all objects of whatever type or size. Even for photons, the concepts of momentum and conservation of momentum are still crucially important even at that scale. (Since they are massless, the momentum of a photon is defined very differently from the momentum of ordinary objects. You will learn about this when you study quantum physics.)
Check Your Understanding Suppose the second, smaller cart had been initially moving to the left. What would the sign of the final velocity have been in this case?
If the smaller cart were rolling at 1.33 m/s to the left, then conservation of momentum gives
Thus, the final velocity is 0.135 m/s to the left.
