Momentum Formulation of Second Law

Newton's Second Law is most commonly stated as F = ma, but Newton himself wrote it in a more powerful form: force equals the rate of change of momentum. Momentum, denoted p, is the product of an object's mass and velocity. When a net force acts on an object, it changes that object's momentum over time. This formulation is not just a rewrite — it is more general, correctly handling situations where mass itself changes, such as a rocket burning fuel. Think of momentum as 'quantity of motion'. A heavy lorry moving slowly can have the same momentum as a small car moving quickly. When a force acts, it acts like a pump pushing momentum into (or out of) an object. The longer the force acts, or the stronger it is, the more the momentum changes. This idea leads directly to the concept of impulse — the total 'dose' of force delivered over a time interval — which equals the total change in momentum. This momentum formulation underpins collision physics, rocket propulsion, and even fundamental particle physics. Understanding it deeply will allow you to solve problems that the simpler F = ma formula cannot handle elegantly, and it reveals the true symmetry built into Newton's laws of motion.

Key takeaways
  • What is momentum?
  • Newton's original statement
  • Recovering F = ma for constant mass
  • Impulse and the impulse–momentum theorem
  • Units and consistency

What is momentum?

Momentum is defined as p = mv, where m is mass (kg) and v is velocity (m/s). It is a vector quantity — it has both magnitude and direction. A 1,000 kg car travelling north at 20 m/s has a momentum of 20,000 kg·m/s directed north. Two objects can have equal momenta but very different masses and speeds.

Newton's original statement

Newton wrote in the Principia that 'the change of motion is proportional to the motive force impressed'. Here, 'motion' meant what we now call momentum. The modern mathematical statement is: F = dp/dt, meaning the net force equals the rate of change of momentum with respect to time. This is the true, general form of the second law.

Recovering F = ma for constant mass

When mass is constant, dp/dt = d(mv)/dt = m(dv/dt) = ma. So F = ma is simply a special case of the momentum formulation. The momentum form is more fundamental because it also works when mass changes — for example, a rocket ejects exhaust gas, reducing its own mass, so you cannot treat m as fixed.

Impulse and the impulse–momentum theorem

Rearranging F = dp/dt gives F dt = dp. Integrating over a time interval gives the impulse–momentum theorem: J = FΔt = Δp. Impulse J is the product of force and the time it acts (for a constant force). This is why airbags save lives: they extend the collision time, reducing the peak force even though Δp (change in momentum) stays the same.

Units and consistency

The unit of momentum is kg·m/s, which is equivalent to the newton-second (N·s), since 1 N = 1 kg·m/s². This equivalence is not a coincidence — it follows directly from J = FΔt = Δp. Checking units: [N][s] = [kg·m/s²][s] = [kg·m/s] ✓. Always check your units when working with momentum problems to catch errors early.

The Key Equation

F⃗net=dp⃗dt=d(mv⃗)dt\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt} = \frac{d(m\vec{v})}{dt}
  • \vec{F}_{\text{net}} — Net force (N (newtons)): The vector sum of all forces acting on the object.
  • \vec{p} — Momentum (kg·m/s): The product of mass and velocity; a vector quantity in the direction of motion.
  • \frac{d\vec{p}}{dt} — Rate of change of momentum (kg·m/s² = N): How quickly the momentum of the object is changing at any instant.
  • m — Mass (kg): The mass of the object (may be variable, e.g. a rocket).
  • \vec{v} — Velocity (m/s): The instantaneous velocity vector of the object.

Worked Example

Problem. A football of mass 0.45 kg is kicked from rest and reaches a velocity of 30 m/s in 0.050 s. Calculate (a) the change in momentum of the ball, and (b) the average net force exerted on the ball during the kick.

Δp=13.5kg ⋅ m/s,Fnet=ΔpΔt=13.50.050=270 N\Delta p = 13.5 \text{kg}\,\cdot\,\text{m/s}, \quad F_{\text{net}} = \frac{\Delta p}{\Delta t} = \frac{13.5}{0.050} = 270 \text{ N}

Answer. The change in momentum is 13.5 kg·m/s, and the average net force on the ball is 270 N.

Figure
Diagram
Impulse–Momentum Theorem: F·Δt = Δpm = 0.45 kgv₁ = 0 m/sp₁ = 0 kg·m/sF = 270 NΔt = 0.05 sJ = F·Δt = 13.5 N·sm = 0.45 kgv₂ = 30 m/sp₂ = 13.5 kg·m/sBeforeAfterp₁=0p₂=13.5Δp = 13.5 kg·m/s = J
Origins · 1 minSir Isaac Newton and how this was discovered
SI
Sir Isaac Newton
1643–1727 · English

Newton formulated the three laws of motion in his landmark work Philosophiæ Naturalis Principia Mathematica (1687). His Second Law, expressed as the proportionality of force to the rate of change of 'quantity of motion' (momentum), forms the cornerstone of classical mechanics. This formulation was more general than F = ma and underpins all of Newtonian dynamics. Newton reportedly developed his key ideas about motion and gravity during 1665–1666 while Cambridge University was closed due to the Great Plague — effectively his most productive period occurred during a lockdown.

  • Newton's Second Law: F = ma — The F = ma form is a special case of the momentum formulation when mass is constant; understanding both deepens your grasp of the second law.
  • Impulse and Momentum Change — Impulse (FΔt = Δp) is the direct practical consequence of the momentum formulation and essential for solving collision and impact problems.
  • Conservation of Momentum — When no external force acts, dp/dt = 0, so total momentum is conserved — this follows directly from the momentum form of the second law.
  • Velocity and Acceleration — Momentum depends on velocity, and its rate of change relates to acceleration; a firm grasp of kinematics supports momentum calculations.