Momentum
Momentum conservation solves exactly the class of problems energy conservation struggles with -- collisions, where kinetic energy is often not conserved but momentum always is. This unit builds the impulse-momentum theorem, distinguishes elastic from inelastic collisions, extends to two dimensions, and introduces center of mass, which resurfaces directly in Unit 8 when rolling objects need to be treated as translating *and* rotating at once.
What you'll learn
- Calculate momentum and impulse for an object or system.
- Apply the impulse-momentum theorem to relate force over time to change in momentum.
- Apply conservation of momentum to isolated systems in 1D and 2D.
- Distinguish elastic from inelastic collisions and identify which quantities are conserved.
- Solve for post-collision velocities using conservation of momentum (and energy, for elastic collisions).
- Calculate center of mass for a system of particles.
- Interpret force-vs-time graphs to determine impulse.
1. Momentum and Impulse
Momentum is \(\vec p = m\vec v\) — a vector that points the same way as velocity. Impulse is the product of a force and the time interval it acts over, \(\vec J = \vec F\Delta t\), and the impulse-momentum theorem states this equals the resulting change in momentum: \(\vec J = \Delta \vec p\).
This relationship explains why airbags, crumple zones, and bent knees when landing all reduce injury: for a given change in momentum \(\Delta p\), stretching the collision over a longer \(\Delta t\) reduces the average force \(F = \Delta p/\Delta t\) needed to produce it.
2. Conservation of Momentum
In an isolated system (no net external force), total momentum is conserved: \(\Sigma \vec p_{before} = \Sigma \vec p_{after}\). This holds for any interaction — a collision, an explosion, objects pushing apart — regardless of what happens to kinetic energy in the process.
3. Elastic Collisions
In an elastic collision, both momentum and kinetic energy are conserved. A useful special case worth memorizing: when two objects of *equal* mass collide elastically and one is initially at rest, the moving object stops completely and the previously stationary object moves off with the first object's original velocity — a full transfer of motion.
4. Inelastic Collisions
In any inelastic collision, momentum is still conserved, but kinetic energy is not — some is converted to heat, sound, or deformation. A perfectly inelastic collision is the extreme case where the objects stick together afterward and move with one common velocity, which produces the maximum possible kinetic energy loss consistent with momentum conservation.
5. Two-Dimensional Collisions
When a collision isn't head-on, momentum is still conserved, but separately in each direction: \(\Sigma p_x\) before equals \(\Sigma p_x\) after, and likewise for \(\Sigma p_y\). Set up x- and y-axes, resolve every velocity into components, and write two independent conservation equations.
6. Center of Mass
The center of mass of a system of particles is the mass-weighted average position: \(x_{cm} = \dfrac{\sum m_i x_i}{\sum m_i}\). It always lies closer to the heavier mass. For an isolated system, the center of mass moves at constant velocity even while individual objects within the system collide, explode apart, or otherwise interact — internal forces can never change the motion of the system's center of mass.
Key equations
- p = mv — Any momentum problem
- J = FΔt = Δp — Relating force/time to momentum change
- Σp(before) = Σp(after) — Conservation of momentum, isolated system
- Elastic collisions: momentum AND KE conserved — Elastic collision problems
- xcm = Σ(mi xi)/Σmi — Multi-object center-of-mass problems