F = ma calculations · Resultant force · Mass vs Weight · Inertial mass
Newton's Second Law is one of the most important relationships in all of physics. It connects three fundamental quantities: force, mass and acceleration.
The equation can be rearranged depending on what quantity you need to find:
What does "directly proportional to force" mean? If you double the resultant force on an object (keeping mass constant), the acceleration doubles. If you triple the force, the acceleration triples. The graph of F against a is a straight line through the origin.
What does "inversely proportional to mass" mean? If you double the mass of an object (keeping force constant), the acceleration is halved. A heavier car is harder to accelerate than a lighter one with the same engine force. The graph of a against m is a curve (a = F/m is a reciprocal relationship).
| Quantity | Symbol | SI Unit | Unit Symbol |
|---|---|---|---|
| Resultant Force | F | Newton | N |
| Mass | m | kilogram | kg |
| Acceleration | a | metres per second squared | m/s² |
Notice that 1 Newton is defined precisely from this equation: 1 N is the force needed to accelerate a 1 kg mass at 1 m/s². So 1 N = 1 kg m/s².
Before you can use F = ma, you must find the resultant force — the single combined force that has the same effect as all the individual forces together.
Forces are vectors, which means they have both size and direction. When adding forces that act along the same line, you simply add or subtract them depending on their directions.
Example — forces in the same direction: A car engine produces 3000 N forwards. Friction and air resistance together produce 800 N backwards.
Example — balanced forces: If the driving force equals friction, the resultant force is zero. By Newton's Second Law, a = F/m = 0/m = 0 m/s². The object moves at constant velocity (or stays stationary). This is Newton's First Law in action.
For forces at angles (e.g. at 90°), you would use Pythagoras' theorem or scale drawings, but at GCSE the forces are almost always along the same line, so simple addition/subtraction applies.
Sign convention tip: Choose one direction as positive (usually right or upwards). Forces in that direction are positive; forces in the opposite direction are negative. The sign of your resultant tells you the direction of acceleration.
Mass and weight are two of the most commonly confused concepts in physics. They are not the same thing, even though in everyday language people often use them interchangeably.
| Location | g (N/kg) |
|---|---|
| Earth's surface | 9.8 (≈ 10 for estimates) |
| Moon's surface | 1.6 |
| Mars's surface | 3.7 |
| Deep space (far from any mass) | ≈ 0 |
Example: A student has a mass of 60 kg. On Earth: W = 60 × 9.8 = 588 N. On the Moon: W = 60 × 1.6 = 96 N. Their mass is still 60 kg in both places.
Notice that weight is a force — it acts downwards towards the centre of the Earth (or whichever planet you're on). When an object is in free fall (no other vertical forces), this weight force causes acceleration downwards at g = 9.8 m/s². You can see this directly from F = ma: W = mg → mg = ma → a = g.
There are actually two ways to define and measure mass, which turn out to give the same result — a fact that puzzled physicists including Einstein.
Why does inertial mass matter? Inertia is the tendency of an object to resist changes in its motion. A shopping trolley full of heavy items has more inertia than an empty one — you need to push much harder to give it the same acceleration, and it's much harder to stop once moving.
How to measure inertial mass experimentally: Apply a known force to an object (e.g. using a stretched elastic band or a newton-meter) and measure the acceleration produced (e.g. using light gates). Then calculate m = F/a. This gives inertial mass without needing gravity at all — you could do this experiment in space!
A key consequence of inertia: Large trucks and lorries are much harder to stop than small cars travelling at the same speed, because they have much greater mass (and therefore inertia). This is why stopping distances for heavy vehicles are much greater — a critical road safety issue.
When tackling F = ma problems in exams, follow these key steps to avoid losing marks:
Step 1 — Identify all forces. Draw a free-body diagram showing all forces acting on the object (driving force, friction, weight, normal reaction, air resistance, tension, etc.).
Step 2 — Find the resultant force. Add forces in the same direction, subtract forces in opposite directions. Be careful with signs and directions.
Step 3 — Apply F = ma. Use the resultant force, not any individual force. Check your units: force in N, mass in kg, acceleration in m/s².
Step 4 — Rearrange if needed. If you need mass: m = F/a. If you need acceleration: a = F/m.
| Given | Find | Equation to Use |
|---|---|---|
| F and m | a | a = F ÷ m |
| F and a | m | m = F ÷ a |
| m and a | F | F = m × a |
| Weight W, location g | mass m | m = W ÷ g |
Always show your working in exams — even if your final answer is wrong, you can earn method marks for correct rearrangement and substitution. Include units at every stage of working and give your final answer to an appropriate number of significant figures (usually 2-3 for GCSE).
Question 1: A resultant force of 500 N acts on an object of mass 50 kg. What is the acceleration of the object?
Question 2: A student has a mass of 65 kg. What is her weight on Earth? (g = 9.8 N/kg)
Question 3: A van of mass 2000 kg accelerates at 3.5 m/s². What is the resultant force acting on it? Enter your answer in N.
Question 4: Which statement best describes inertial mass?
Question 5: A bicycle and rider have a combined mass of 80 kg. The rider pedals with a force of 200 N. Friction produces a resistive force of 40 N. Calculate the acceleration. Enter your answer in m/s².
Challenge 1: A rocket of mass 8000 kg produces a thrust of 120 000 N. The gravitational force (weight) on the rocket is 78 400 N downwards. Air resistance at the moment of launch is negligible. Calculate: (a) the resultant force on the rocket, and (b) its initial acceleration. (g = 9.8 N/kg)
Challenge 2: A skydiver of mass 70 kg jumps from a plane. At one moment during the fall, air resistance is 420 N. Calculate her acceleration at that moment. (g = 9.8 N/kg)
Challenge 3 (Extended): A student investigates inertial mass by applying different forces to a trolley and measuring accelerations. The results are:
| Force (N) | Acceleration (m/s²) | m = F/a (kg) |
|---|---|---|
| 2.0 | 0.80 | ? |
| 4.0 | 1.62 | ? |
| 6.0 | 2.38 | ? |
(a) Calculate the inertial mass for each row. (b) Explain why the three values are slightly different. (c) Estimate the best value for the trolley's inertial mass.
Challenge 4 (6-mark style): A car of mass 1500 kg is travelling at 30 m/s. The driver applies the brakes, producing a braking force of 9000 N. Explain, using Newton's Second Law, what happens to the car, and calculate how long it takes to stop. Use appropriate equations and show all working.