Linear momentum measures the "quantity of motion" of an object. Defined as p = mv, it's a VECTOR — meaning it has both magnitude and direction. A heavier object or a faster object has more momentum. Momentum is used to model collisions (where forces between objects are huge) and explosions (where internal forces push objects apart).
Key Properties
Formula
p = mv. Linear (not squared).
Type
Vector. Same direction as velocity.
Units
kg·m/s (= N·s, same as impulse)
Total p of System
Vector sum of all individual momenta.
Example
Two carts on a track
Cart A (2 kg) moves east at 4 m/s. Cart B (3 kg) moves west at 2 m/s. Taking east as positive:
p_A = 2 · (+4) = +8 kg·m/s (eastward)
p_B = 3 · (−2) = −6 kg·m/s (westward)
Total system momentum: p_A + p_B = +2 kg·m/s (eastward)
Even though Cart B is heavier, A's higher speed gives it more momentum, and the system has a net eastward momentum.
Topic 4.2
Change in Momentum and Impulse
How forces transfer momentum through time
The Big Idea
An impulse (J = F·Δt) is what you get when a force acts for some time. The impulse-momentum theorem says J = Δp — the impulse you deliver equals the change in momentum. This is the secret behind airbags, crumple zones, and bent knees on landing: extend Δt, reduce F, same total Δp.
Key Equations & Graphs
Impulse
J = F_avg · Δt. Units: N·s.
Impulse-Momentum Theorem
J = Δp = p_f − p_i
F-t Graph
Area under curve = impulse
p-t Graph
Slope = net force (F = Δp/Δt)
Example
Catching a baseball
A 0.15 kg baseball is moving at 30 m/s when you catch it. The ball stops in your glove. The impulse on the ball is:
If your hand stops the ball in 0.05 s (rigid hand): F_avg = J/Δt = −4.5/0.05 = −90 N (90 N back on the ball; ball pushes 90 N forward on your hand).
If your hand "gives" and stops it over 0.2 s (soft catch): F_avg = −4.5/0.2 = −22.5 N — only one-fourth the force. Same impulse, gentler hand.
Topic 4.3
Conservation of Linear Momentum
When no external force acts, total momentum is constant
The Big Idea
For a closed system with no net external force, total momentum stays constant: p_i,total = p_f,total. This is one of the most powerful tools in physics. Internal forces between objects come in Newton's-third-law pairs that cancel, so they can never change the system's total momentum. This makes collisions solvable even when the internal forces are messy.
How to Use It
Choose your system to include all interacting objects. That way internal forces cancel.
Identify before and after velocities (with signs for direction).
Write p_i = p_f: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'.
Solve for the unknown, typically a final velocity.
Example
A skateboarder throws a ball
A 50 kg skateboarder is at rest on frictionless ice. They throw a 2 kg ball east at 5 m/s. How fast does the skater move backward?
Initial momentum of system (skater + ball): 0 (both at rest).
0 = (2)(+5) + (50)(v_skater) → v_skater = −0.2 m/s
The skater drifts west at 0.2 m/s — twenty-five times slower than the ball, because they're twenty-five times heavier. Internal forces (skater pushing ball) can't change the system's zero total momentum.
Topic 4.4
Elastic and Inelastic Collisions
Momentum always conserved; KE sometimes
The Big Idea
In EVERY collision (with no net external force), momentum is conserved. The difference between elastic and inelastic collisions is about KINETIC ENERGY:
Elastic: KE is conserved (KE_i = KE_f). Both momentum AND kinetic energy survive.
Inelastic: KE decreases. Some becomes heat, sound, or deformation energy. Momentum is still conserved.
Perfectly inelastic: objects stick together. MAXIMUM KE is lost.
Identifying the Type
Elastic
KE_i = KE_f. Billiard balls (approx).
Inelastic
KE_f < KE_i. Most real collisions.
Perfectly Inelastic
Stick together. Max KE lost.
Both Always Conserve
Total system momentum.
Example
Two carts stick together
A 1 kg cart moves east at 4 m/s and collides with a 3 kg cart at rest. They stick together (perfectly inelastic).
Find v_f: (1)(4) + (3)(0) = (4)(v_f) → v_f = 1 m/s east
KE before: ½(1)(4)² + 0 = 8 J
KE after: ½(4)(1)² = 2 J
KE lost = 6 J (75% of the original KE!). Where did it go? Heat from friction between the carts, sound, and energy used to deform them. Momentum is conserved (4 kg·m/s before and after), but most KE is gone.
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How to use this visual review
Take 1–2 minutes per slide. Read the Big Idea first, then the definitions, then the worked example. Four slides total — one per topic.
Use the topic pills to jump to any topic, or use the arrow keys to step through them in order.
Unit 4 is short — this works perfectly as a last-minute refresher before the exam or a quick review after watching a lecture.