4.1 Linear Momentum

Defining Linear Momentum

Momentum (p) is a vector quantity defined as p = mv, the product of an object's mass and velocity. It points in the same direction as velocity. Momentum describes how hard it is to stop a moving object — a heavy truck moving slowly can have the same momentum as a light car moving fast. Units are kg·m/s.

Momentum of a System

For a system of particles, total momentum is the vector sum of each object's individual momentum: p_total = Σmᵢvᵢ. Because momentum is a vector, direction matters — momenta in opposite directions can partially or fully cancel. This concept sets up conservation of momentum and collision analysis.

Momentum and Force (Calculus View)

Newton's second law can be written as F_net = dp/dt, the rate of change of momentum. When mass is constant, this reduces to F = ma. This more general form applies even when mass changes (like a rocket losing fuel), making momentum a more fundamental quantity than force alone.

Key terms

Linear momentum (p)

p = mv; a vector quantity in the direction of velocity, measured in kg·m/s.

Momentum of a system

The vector sum of all individual momenta: p_total = Σmᵢvᵢ.

Generalized Newton's second law

F_net = dp/dt — net force equals the rate of change of momentum, valid even with changing mass.

Why momentum is a vector

Because it depends on velocity's direction; momenta can add or cancel depending on direction.

Momentum vs. force insight

Momentum accounts for both mass and motion, making it useful for analyzing collisions and systems where force alone is hard to apply.

Try an AP-style question

A 3 kg object has velocity (2î − 4ĵ) m/s. What is its momentum?

1. √20 kg·m/s
2. (6î − 12ĵ) kg·m/s
3. (2î − 4ĵ) kg·m/s
4. (5î − 7ĵ) kg·m/s