Newton's Second Law: F = ma
Newton's second law equation states that the net force F acting on an object of constant mass m equals the product of that mass and the resulting acceleration a: F = ma.
The equation
| Symbol | Name | SI Unit | Description |
|---|---|---|---|
| F | Net force | newton (N) = kg·m/s² | Vector sum of all forces acting on the object |
| m | Mass | kilogram (kg) | Inertial mass, assumed constant |
| a | Acceleration | m/s² | Rate of change of velocity |
Conditions of validity
- Holds exactly only in inertial (non-accelerating) reference frames
- Requires constant mass — for variable-mass systems, use the general form F = dp/dt
- Valid for speeds much less than the speed of light (v « c); at relativistic speeds, replace with F = d(γmv)/dt
- Assumes classical, non-quantum scales — for quantum systems, use Ehrenfest's theorem as the analogous statement about expectation values
Derivation
Newton's second law in its most general form is stated in terms of momentum:
Step 1. Momentum is defined as p = mv.
Step 2. Substitute into the general law: F = d(mv)/dt.
Step 3. Apply the product rule for differentiation: F = m(dv/dt) + v(dm/dt).
Step 4. For a system of constant mass, dm/dt = 0, so the second term vanishes: F = m(dv/dt).
Step 5. By definition, acceleration a = dv/dt. Substituting gives the familiar form:
This derivation makes explicit why F = ma fails for variable-mass systems such as rockets: Step 4's assumption dm/dt = 0 is false for a rocket burning fuel, and the discarded term v(dm/dt) becomes physically significant, forming the basis of the Tsiolkovsky rocket equation.
Worked examples
Example 1 — solving for force. A 1,500 kg car accelerates at 2.5 m/s². Find the net force required.
F = ma = 1,500 × 2.5 = 3,750 N
Example 2 — solving for acceleration. A net force of 45 N acts on a 9 kg object. Find its acceleration.
a = F/m = 45 / 9 = 5 m/s²
Example 3 — solving for mass. A net force of 200 N produces an acceleration of 4 m/s². Find the mass.
m = F/a = 200 / 4 = 50 kg
Interactive calculator
Enter any two values to solve for the third. Leave the value you want solved for at 0.
Historical note
Isaac Newton (1642–1727) stated the second law in Book I of the Philosophiæ Naturalis Principia Mathematica, published in 1687. Newton's original phrasing referred to "the change of motion" (momentum) being proportional to impressed force, not to the algebraic F = ma form used today. The explicit algebraic notation is generally attributed to Leonhard Euler's 18th-century reformulation of mechanics.
Related equations
- Linear Momentum (p = mv)
- Work-Energy Theorem [PENDING FULL ENTRY]
- Newton's Third Law
Primary Sources
- Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica, Book I, Law II. Public domain.
- Goldstein, H., Poole, C., Safko, J. (2002). Classical Mechanics, 3rd ed. Addison Wesley, Ch. 1.
- NIST SI Brochure, 9th ed. — definition of the newton (N). nist.gov