Newton's Second Law
Newton's second law states that the net force acting on an object equals the rate of change of its momentum with time; for an object of constant mass, this reduces to F = ma, where F is the net force, m is the mass, and a is the resulting acceleration.
Curious
Imagine pushing an empty shopping cart versus pushing one loaded with bricks. Same push, same force from your arms — but the empty cart shoots forward while the loaded one barely budges. That difference is entirely captured by one idea: acceleration depends on how much force you apply and how much mass is resisting it.
Newton's second law puts a number on this. It says the acceleration of an object is directly proportional to the net force acting on it, and inversely proportional to its mass. Double the force, and you double the acceleration. Double the mass while keeping the force the same, and you halve the acceleration.
This single relationship is why rockets need to burn away fuel mass to accelerate efficiently, why a bowling ball is harder to speed up than a tennis ball with the same push, and why cars with more powerful engines accelerate faster than heavier trucks with the same engine.
The law only works cleanly for the net force — if you push a box left while friction pushes it right, it's the leftover, unbalanced force that produces acceleration. If forces are perfectly balanced, there's no acceleration at all, no matter how hard everyone is pushing.
Exploring
In its most common working form, the law is written:
| Symbol | Quantity | SI Unit |
|---|---|---|
| F | Net force (vector sum of all forces) | newton (N) = kg·m/s² |
| m | Mass (inertial mass, assumed constant) | kilogram (kg) |
| a | Acceleration | m/s² |
This is the constant-mass special case of Newton's more general and more fundamental statement, which defines force as the rate of change of momentum:
When mass doesn't change, dp/dt = m(dv/dt) = ma, and the familiar F = ma falls out directly. The momentum form matters when mass is not constant — a rocket burning fuel, a raindrop accreting mass as it falls, or a conveyor belt loading material — because in those cases F = ma alone gives the wrong answer, while F = dp/dt remains correct.
Force and acceleration are both vectors, and the equation applies component-by-component: F_x = ma_x, F_y = ma_y, F_z = ma_z. This is why problems are usually solved by resolving all forces into perpendicular components before applying the law.
Worked example 1: A 1,200 kg car accelerates from rest to 27 m/s (about 97 km/h) in 9 seconds. Its acceleration is a = Δv/Δt = 27/9 = 3 m/s². The net forward force required is F = ma = 1,200 × 3 = 3,600 N.
Worked example 2: A 0.5 kg ball is hit with a force of 20 N to the right while air resistance exerts 4 N to the left. Net force = 20 − 4 = 16 N. Acceleration = F/m = 16/0.5 = 32 m/s².
Deep Dive
Newton's original statement in the Principia is phrased in terms of "motion" (his term for what is now called momentum, p = mv) rather than acceleration. The modern F = ma form became standard only after Euler's 18th-century reformulation of mechanics in explicitly algebraic terms; Newton himself never wrote the equation F = ma in his published work.
The fully general, relativistically-consistent form of the second law is:
where p is relativistic momentum, p = γmv, with γ = 1/√(1 − v²/c²). In the low-velocity limit (v ≪ c), γ → 1 and mass is effectively constant, recovering F = ma. At relativistic speeds, F = ma is no longer valid even in principle — force and acceleration are not generally parallel, and the effective inertial resistance to acceleration increases with velocity.
Derivation of F = ma from F = dp/dt (constant mass case):
Start from the momentum form: F = dp/dt. Since p = mv and m is constant, differentiate:
This step is only valid when dm/dt = 0. If mass varies with time (as in rocket propulsion), the product rule gives F = m(dv/dt) + v(dm/dt), and the extra term v(dm/dt) must be accounted for explicitly — this is the basis of the Tsiolkovsky rocket equation.
Conditions of validity: Newton's second law, in the form F = ma, holds exactly only in inertial reference frames, for speeds much less than the speed of light, and for objects large enough that quantum effects are negligible. In non-inertial (accelerating) reference frames, fictitious forces (centrifugal, Coriolis, Euler) must be added to the real forces for the law to hold in that frame's coordinates. In quantum mechanics, an analogous relation (Ehrenfest's theorem) shows that the expectation values of position and momentum obey a form of Newton's second law, but individual particles do not follow deterministic trajectories.
Historical note: Isaac Newton (1642–1727) presented the three laws of motion in Book I of the Principia Mathematica, published in Latin in 1687 with the support of Edmond Halley. The Principia unified terrestrial and celestial mechanics under a single set of laws, showing that the same force law governing a falling apple also governs the orbit of the Moon.
Concept Relationship Map
Requires: force, inertia (First Law), mass
Enables: Newton's Third Law (as a paired system), momentum conservation, work-energy theorem
Commonly confused with: Newton's First Law (which describes zero net force, not the general case)
Sources
- Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica, Book I, Axioms, Law II. Public domain.
- Goldstein, H., Poole, C., Safko, J. (2002). Classical Mechanics, 3rd ed. Addison Wesley, Ch. 1.
- Feynman, R.P. et al. The Feynman Lectures on Physics, Vol. I, Ch. 9. feynmanlectures.caltech.edu
- NIST. SI Brochure, 9th ed. — definition of the newton. nist.gov