AP® Physics C: Mechanics · Unit 4 · Topic 4.1
Momentum of a System
The total momentum of a system is the vector sum of every object's momentum inside it, equal to the system's total mass times the velocity of its center of mass. Internal forces, always paired by Newton's third law, can never change that total.
Momentum is p = mv, a vector, so when a system has several objects you add their momenta as vectors, not their speeds. Two carts moving toward each other can have momenta that partly or fully cancel even though both are moving fast. That total is also equal to M_total times v_cm, the velocity of the system's center of mass. That's not a coincidence: differentiate the center of mass position, M_total·x_cm = sum of mi·xi, with respect to time and you get exactly that relationship.
Why does slamming two carts together not touch the total? Every force the carts exert on each other during the collision is internal to the system, and Newton's third law guarantees each such force is matched by an equal and opposite partner. Add up all the internal forces at any instant and they cancel exactly, piece by piece. Since force is the rate of change of momentum, forces that sum to zero produce no net change in total momentum. Only an external force — friction from the ground, a push from outside the system — can shift the total.
This is the real content behind Newton's second law applied to a system: F_net,external = dp_total/dt. When no external force acts, p_total is constant, which is exactly why the center of mass of an isolated system moves in a straight line at constant velocity even while its pieces crash, bounce, or stick together. Everything in the rest of the unit — impulse, collisions, explosions — is just this one idea applied to specific situations.
Key terms
- Momentum
- The vector quantity p = mv, describing how much motion an object carries.
- System
- A chosen collection of objects treated as one unit, with momentum added across all of them.
- Internal force
- A force exerted between two objects that are both inside the chosen system.
- Center of mass velocity
- The velocity of the point representing the system's average mass position, v_cm = p_total / M_total.
- External force
- A force on the system exerted by something outside it, the only kind that can change total momentum.
Watch it explained
- How it worksHow does a system's momentum equal its total mass times its center-of-mass velocity?
- Worked exampleFind the total momentum of a 4 kg cart moving 3 m/s east and a 6 kg cart moving 2 m/s west.
- Common mistakeWhy don't internal forces inside a system ever change the system's total momentum?
- Ties togetherHow does system momentum tie back to the way the center of mass moves?
Check yourself
End of topic.