Topic 4.1 Linear Momentum: define linear momentum as the product of mass and velocity, treat it as a vector, and relate the net force to its rate of change.

What this topic is asking

The College Board (Topic 4.1) wants you to define linear momentum as the product of mass and velocity, to treat it as a vector, and to relate the net force to its rate of change. Momentum is the quantity that conservation laws and collision analysis are built on, and recognizing that it is a vector, with direction as well as magnitude, is essential to everything in Unit 4.

Defining momentum

Momentum measures the "quantity of motion": a heavy, fast object has large momentum and is hard to stop. Because it is a vector, two objects moving in opposite directions at equal speeds have momenta that cancel when summed. This vector character is what distinguishes momentum problems from energy problems: kinetic energy is a scalar and adds as numbers, while momentum must be added by components, respecting direction. Note also that momentum is linear in speed, so doubling the speed doubles the momentum (but quadruples the kinetic energy).

Momentum of a system

For a system of several objects, the total momentum is the vector sum of the individual momenta:

A beautiful result connects this to the center of mass: since $\vec{v}_{cm} = \dfrac{\sum m_i\vec{v}_i}{M}$, the total momentum equals the total mass times the center-of-mass velocity,

So the whole system moves, momentum-wise, as if its entire mass were concentrated at the center of mass. This is why momentum and the center-of-mass concept from Unit 2 are two views of the same physics, and it is the foundation of momentum conservation.

Newton's second law as the rate of change of momentum

The deepest statement of Newton's second law is in terms of momentum:

The net external force equals the time rate of change of the momentum. When the mass is constant this is $\vec{F}_{net} = m\dfrac{d\vec{v}}{dt} = m\vec{a}$, the familiar form. But the momentum form is more general: it correctly handles systems whose mass changes (a rocket ejecting fuel, a cart gathering sand) where $m\vec{a}$ alone would be wrong. It also leads directly to the impulse-momentum theorem in the next topic, since integrating the force over time gives the change in momentum.

Try this

Q1. A $0.15$ kg baseball moves at $40$ m/s. Calculate its momentum. [2 points]

• Cue. $p = mv = (0.15)(40) = 6.0$ kg m/s in the direction of motion.

Q2. State the general form of Newton's second law in terms of momentum and explain when it differs from $\vec{F} = m\vec{a}$. [2 points]

• Cue. $\vec{F}_{net} = d\vec{p}/dt$; it differs from $m\vec{a}$ when the mass changes, such as a rocket losing fuel.