Newton's second law of motion

Newton's second law says that the acceleration of an object is proportional to the resultant force acting on it and inversely proportional to its mass. As an equation, F = ma: resultant force (N) = mass (kg) × acceleration (m/s²). A bigger force gives a bigger acceleration, and a heavier object needs a bigger force for the same acceleration. One newton is the force that gives a 1 kg mass an acceleration of 1 m/s².

Newton's second law of motion

The word that matters most is resultant. F is not just one push or pull: it is what is left after all the forces on the object have been added up, taking their directions into account. If the forces balance, the resultant is zero and there is no acceleration, which is Newton's first law. Newton's second law tells you what happens when they do not balance.

Finding the resultant force

Draw a free body diagram: the object, with an arrow for each force acting on it. Add the forces that point the same way, and subtract the ones that point the opposite way.

Newton's second law examples

Example 1: finding the force

A 1200 kg car accelerates at 2.5 m/s². What resultant force acts on it?

Example 2: from two forces

The box in the diagram has a mass of 60 kg. It is pushed with a force of 500 N against 200 N of friction. Find its acceleration.

Resultant force = 500 − 200 = 300 N to the right. The weight and normal contact force balance, so they add nothing.

Example 3: finding the mass

A resultant force of 24 N gives an object an acceleration of 3 m/s². Find its mass.

Example 4: braking

A 1500 kg car slows from 20 m/s to rest in 4 s. Find the braking force.

First the acceleration: a = (0 − 20) ÷ 4 = −5 m/s². Then F = ma = 1500 × (−5) = −7500 N. The braking force is 7500 N, acting backwards, against the motion.

Example 5: a lift accelerating upwards

An 800 kg lift accelerates upwards at 1.5 m/s². How hard must the cable pull on it? Use g = 9.8 N/kg.

Two forces act: the tension T upwards and the weight, 800 × 9.8 = 7840 N, downwards. The resultant is T − 7840, and it must equal ma:

The cable has to hold the weight and supply the extra force for the acceleration. This is why you feel heavier when a lift starts going up.

Now try one. Use F = ma, then press Mark to check your answer in the course.

Weight is F = ma too

When an object falls freely, the only force is its weight, and its acceleration is g. Putting these into F = ma gives W = mg, the weight equation. That is why every object, heavy or light, falls with the same acceleration if air resistance is negligible: a heavier object has a bigger weight, but also a bigger mass to accelerate, and the two cancel.

Inertial mass

AQA's GCSE specification asks Higher tier students to explain that inertial mass is a measure of how difficult it is to change the velocity of an object, and that it is defined as the ratio of force over acceleration, m = F ÷ a. An object with twice the inertial mass needs twice the force for the same acceleration. This is the same as the mass that appears in weight, which is why F = ma and W = mg use the same m.

The required practical: force and acceleration

In the GCSE required practical, a trolley on a runway is pulled by a hanging mass. To vary the force while keeping the total mass constant, move masses from the trolley to the hanger. Light gates or a video measure the acceleration. Plotting acceleration against force gives a straight line through the origin, showing a ∝ F. Repeating with different trolley masses shows that a larger mass gives a smaller acceleration for the same force. The Practical Data Lab helps you plot and analyse these results.

Newton's second law and momentum

Newton's own version of the second law says that force equals the rate of change of momentum, F = Δ(mv) ÷ Δt. When the mass stays the same, Δ(mv) = mΔv, and Δv ÷ Δt is the acceleration, so this becomes F = ma. The momentum form also works when the mass changes, such as a rocket burning fuel. Our guide to momentum uses it for collisions and impacts.

Common mistakes

• Using one force instead of the resultant. Subtract friction, drag or other opposing forces first.
• Mass in grams. 500 g is 0.5 kg.
• Using weight as the mass. A weight in newtons must be divided by g to get the mass in kg.
• Forgetting the weight in vertical problems. In a lift or a rocket, the resultant is the upward force minus the weight.
• Thinking a moving object needs a resultant force. A resultant force changes velocity. Constant velocity needs zero resultant force.

Is F = ma on the equation sheet?

Yes. AQA's GCSE Physics equations sheet, checked in November 2026, gives resultant force = mass × acceleration, F = ma, and AQA's specification lists it as an equation to recall and apply. The sheet also gives W = mg. Other boards publish their own sheets, so check yours. The Force Calculator works out F, m or a from the other two.

Practise forces in a short timed paper

GCSE Physics Higher and A-level Physics: Year 1 each give you a short timed paper every day, with 8 questions worth 25 to 35 marks across the course. The Laws of Motion course covers all three of Newton's laws with interactive examples.

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