Newton’s Second Law: F = ma

Newton’s second law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass, a = F_net ÷ m, usually written F_net = ma; more generally, the net force equals the rate of change of momentum, F = Δp ÷ Δt. In brief: the acceleration has the same direction as the net force. The unit newton is defined by this law: 1 N = 1 kg·m/s². The law is applied separately to each direction, ΣF_x = ma_x and ΣF_y = ma_y.

This law is the key equation of classical mechanics. It allows the motion of an object to be predicted from the forces acting on it. This guide states the law and shows how to use it with free-body diagrams, compares the effects of force and mass in a table, works through problems with several forces, and corrects common mistakes.

What does the second law say?

The law relates three quantities: the net force F_net (the vector sum of all forces, in newtons), the mass m (in kilograms) and the acceleration a (in m/s²):

F_net = ma, or a = F_net ÷ m.

• For a given mass, doubling the net force doubles the acceleration.
• For a given force, doubling the mass halves the acceleration.
• The acceleration is in the direction of the net force.
• If F_net = 0, a = 0, which is the first law.

In the more general form F = dp/dt, with momentum p = mv, the law also applies to systems whose mass changes, such as a rocket that loses fuel. The law holds in inertial frames and for speeds much less than the speed of light.

How is it used?

1. Choose the object and draw a free-body diagram with all the forces acting on it.
2. Choose axes (x along the direction of the acceleration, if known).
3. Resolve forces into components, and add them: ΣF_x and ΣF_y.
4. Write ΣF = ma in each direction and solve for the unknown (acceleration, force, mass).
5. Check the units and the plausibility of the answer.

Weight, W = mg, is a special case: the only force in free fall is the weight, and the acceleration is g.

Effects of force and mass compared

Worked example: applying F = ma

1. Simple push. A 2.0 kg block is pushed with a horizontal force of 10 N, and a friction force of 4.0 N opposes the motion. F_net = 10 − 4.0 = 6.0 N, so a = 6.0 ÷ 2.0 = 3.0 m/s².
2. Finding the force. A 1,200 kg car accelerates from 0 to 27 m/s in 9.0 s. a = 27 ÷ 9.0 = 3.0 m/s², so F_net = 1,200 × 3.0 = 3,600 N.
3. Finding the mass. A net force of 30 N gives an unknown cart an acceleration of 1.5 m/s². m = F ÷ a = 30 ÷ 1.5 = 20 kg.
4. Force at an angle. A 5.0 kg box on a smooth floor is pulled by a force of 20 N at 30° above the horizontal. The horizontal component is 20 × cos 30° = 17.3 N, so a = 17.3 ÷ 5.0 = 3.5 m/s². The vertical component, 20 × sin 30° = 10 N, reduces the normal force: N = 5.0 × 9.8 − 10 = 39 N.
5. Falling with air resistance. A 0.50 kg ball falls with air resistance of 1.5 N. F_net = 0.50 × 9.8 − 1.5 = 3.4 N downward, so a = 3.4 ÷ 0.50 = 6.8 m/s².

How did Newton state the law?

In the Principia (1687), Newton wrote that “the change of motion is proportional to the motive force impressed”, where “motion” meant momentum. The form F = ma was introduced later by Leonhard Euler in the 1750s. The law is the basis of classical mechanics, which was used to calculate the orbits of the planets and, in the 20th century, spacecraft trajectories. At speeds near that of light, relativistic mechanics is required.

Which mistakes are common?

Mistake 1: Using a single force instead of the net force

Acceleration depends on the vector sum of all forces.

Mistake 2: Thinking that force is proportional to velocity

Force is proportional to acceleration. A constant velocity needs no net force.

Mistake 3: Using weight instead of mass

In F = ma, m is the mass in kg, not the weight in N.

Mistake 4: Ignoring the direction

Force and acceleration are vectors. Use components with signs.