Physics I: Mechanics › Momentum › full formula sheet

p = mv

Say it: “momentum equals mass times velocity — the quantity of motion”

Linear momentum

The true measure of 'how much motion' — why a slow truck outranks a fast bullet, and the quantity collisions conserve.

p is momentum in kg·m/s, m mass (kg), v velocity (m/s) — with its sign. Momentum is a vector: direction matters.

Where it comes from

A 0.01 kg bullet at 300 m/s and a 2000 kg truck at 1.5 m/s — which is harder to stop? The bullet is 200× faster, yet the truck wins: 3000 kg·m/s of motion vs the bullet's 3. Speed alone lies; mass × velocity tells the truth.

Before reading on: a 1000 kg car at 30 m/s vs a 2000 kg truck at 10 m/s — which carries more momentum? Which would you rather be hit by (if you had to choose)?

The car: 30,000 vs 20,000 kg·m/s. Momentum multiplies mass and velocity into one number — the “quantity of motion” that determines how hard something is to stop and what happens in collisions:

p = mvmass times velocity — the quantity of motion, and a vectorSay it: “momentum equals mass times velocity”

Newton actually wrote his second law in terms of momentum: F = dp/dt. Force is the rate of change of momentum — which is why momentum, not velocity, is the star of collisions.

Derivation

Momentum is a definition, not a theorem — but Newton's second law reveals why it's the right definition:

F
=
ma = m(dv/dt)
Step 1 — Newton. The familiar second law, with acceleration written as dv/dt.
=
d(mv)/dt
Step 2 — constant mass. m slides inside the derivative (it doesn't change with time).
F
=
dp/dt
Step 3 — name it. Define p = mv: force is the rate of change of momentum. This is the form Newton actually wrote — and the doorway to impulse and conservation. ∎

How to use it

The procedure, every time:

  1. Fix a positive direction first. “Right is positive” — write it down. Every velocity then gets a sign.
  2. Keep the sign on v. p = mv with signed v: motion against your positive direction gives negative momentum. Dropping signs destroys collision problems.
  3. Multiply. p = mv in kg·m/s. Big mass × small speed can beat small mass × big speed.
  4. Compare momenta, not speeds. “Harder to stop” and “hits harder” are momentum questions.
  5. Remember it's a vector. Two objects with equal speeds in opposite directions have momenta +p and −p — their total is zero.
Common mistake: using speed (unsigned) where velocity (signed) belongs. In collisions the signs do half the work — p = +12 and p = −12 are very different momenta.

Worked examples

Four problems, easiest first. Signs are half the lesson.

Example 1 — basic: m = 3 kg, v = +4 m/s

  1. Multiply. p = 3 × 4 = 12 kg·m/s.
  2. Sign. Positive — moving in the +x direction.
Common mistake: writing the unit as “kg/m/s” or just “12”. Momentum's unit is kg·m/s — say it as “kilogram meters per second.”
Your turn — m = 5 kg, v = +2 m/s. p = ?

Answer: 10 kg·m/s. 5 × 2 = 10.

Example 2 — mass vs speed: truck 2000 kg at 10 m/s vs car 1000 kg at 30 m/s

  1. Truck. p = 2000 × 10 = 20,000 kg·m/s.
  2. Car. p = 1000 × 30 = 30,000 kg·m/s.
  3. Verdict. The car carries more momentum — it's harder to stop despite half the mass. Speed can beat mass.
Common mistake: “the truck is heavier so it has more momentum.” Heavier at equal speed — yes. But momentum multiplies; always compute.
Your turn — bike + rider 100 kg at 5 m/s vs runner 80 kg at 6 m/s. More momentum?

Answer: the bike, 500 vs 480 kg·m/s. 100×5 = 500; 80×6 = 480.

Example 3 — direction: m = 2 kg, v = −7.5 m/s (moving in −x)

  1. Multiply with the sign. p = 2 × (−7.5) = −15 kg·m/s.
  2. Read it. 15 kg·m/s of motion in the negative direction. The minus is information, not an error.
Common mistake: reporting 15 kg·m/s and losing the direction — then adding it to a +15 as 30 instead of canceling to 0 in a collision problem.
Your turn — m = 2 kg, p = −8 kg·m/s. v = ?

Answer: −4 m/s. v = p/m = −8/2 = −4 m/s (moving in −x).

Example 4 — solve for v: p = 50 kg·m/s, m = 2 kg

  1. Rearrange. v = p/m.
  2. Compute. v = 50/2 = 25 m/s (in the + direction).
Common mistake: v = m/p = 0.04 m/s — flipping the fraction. Check: p = mv must give back 50: 2 × 25 = 50 ✓.
Your turn — p = 120 kg·m/s, m = 4 kg. v = ?

Answer: 30 m/s. 120/4 = 30 m/s.

Memorization tips

  • Say it aloud: “momentum equals mass times velocity — signed velocity.”
  • Signs first: declare the positive direction before any collision problem. It's half the setup.
  • Multiply, don't eyeball: heavy-slow vs light-fast is always a computation, never a guess.
  • Vector, not scalar: +p and −p cancel. Momentum adds like arrows, not like money.
  • Unit chant: kilogram-meters-per-second. If your unit isn't that, recheck.
  • Newton's real law: F = dp/dt. Force changes momentum — the seed of impulse and conservation.

Final challenge

Five mixed questions — signs, comparisons, and rearrangements. Score 5/5 and momentum is yours.

← Back to the Physics I formula sheet

How to learn a formula here

  1. Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
  2. Work the examples with the answers covered, then uncover one step at a time and compare.
  3. Finish with the final challenge — five mixed questions including the classic traps.
  4. Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.

Frequently asked questions

Why is momentum mv instead of just v?

Because stopping power depends on both: a truck at 10 m/s is far harder to stop than a bullet at 300 m/s. Mass × velocity is the combination that actually predicts collision outcomes and stopping difficulty.

Is momentum a vector or a scalar?

A vector — it inherits velocity's direction. The sign (in 1D) or arrow (in 2D) matters: opposite momenta cancel in totals, which is the entire basis of conservation.

What's the unit kg·m/s in terms of force?

Since F = dp/dt, momentum = force × time: kg·m/s = N·s. Same quantity, two costumes — the N·s form appears in the impulse page.

Can momentum be zero for a moving object?

Only if m = 0 (a photon — special relativity territory) or v = 0. Anything with mass that's moving carries momentum, full stop.

How is momentum different from kinetic energy?

Momentum is mv (vector, linear in v); kinetic energy is ½mv² (scalar, squared). Collisions conserve momentum always; kinetic energy only sometimes. They answer different questions.

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