The resultant force on an object equals its mass multiplied by its acceleration: F = ma. The force in this equation is always the resultant (net) force, and the acceleration points in the same direction as that force.
This is the central calculation of forces and momentum. It builds on calculating a resultant force, and the acceleration can be checked against the gradient ideas in finding acceleration from a gradient.
Why does the resultant force matter?
An object accelerates only when the forces on it are unbalanced. A car with an engine force of 3000 N does not accelerate by 3000 N worth if friction and air resistance take some of it back. Only the leftover, the resultant, makes the car speed up.
A larger mass needs a larger resultant force for the same acceleration. That is why an empty trolley speeds up more quickly than a loaded one under the same push.
How do you use F = ma?
- Find the resultant force along the direction of motion: add and subtract as in the previous lesson.
- Convert the mass to kilograms if it is given in grams.
- Rearrange the equation to the quantity you need: a = F/m, F = ma or m = F/a.
- Substitute and calculate, then state the unit.
- Check sense: a bigger resultant force should give a bigger acceleration.
Worked example
Invented situation: a car of mass 1200 kg has an engine driving force of 3000 N. The total friction and air resistance is 900 N. Find the acceleration, and the speed after 4.0 s from rest.
Step 1, resultant force: forwards is positive. 3000 − 900 = 2100 N forwards.
Step 2, mass: already 1200 kg.
Step 3, rearrange: a = F/m.
Step 4, substitute: a = 2100 ÷ 1200 = 1.75 m/s², in the forward direction.
Step 5, speed after 4.0 s: v = u + at, with u = 0, so v = 1.75 × 4.0 = 7.0 m/s.
Check: F = ma → 1200 × 1.75 = 2100 N. This matches the resultant force found in step 1.
The mistake to watch for
The usual slip is to put the driving force straight into the equation.
Mistaken working: a = 3000 ÷ 1200 = 2.5 m/s².
The student used the engine force instead of the resultant. The 900 N of resistance was ignored, so the acceleration is too large.
The correction is to write “resultant = …” as its own line before using F = ma. The fix takes ten seconds and removes the most common error. Check the answer the other way as well: 2.5 m/s² would need a resultant of 3000 N, which is not what the question gave.
Check yourself
Try these, then open each answer.
1. A trolley of mass 4.0 kg has a resultant force of 10 N. Find its acceleration.
Show answer
a = F/m = 10 ÷ 4.0 = 2.5 m/s². Check: 4.0 × 2.5 = 10 N.
2. A resultant force of 35 N gives an object an acceleration of 0.50 m/s². Find its mass.
Show answer
m = F/a = 35 ÷ 0.50 = 70 kg. Check: 70 × 0.50 = 35 N.
3. A ball of mass 600 g is kicked and accelerates at 15 m/s². Find the resultant force.
Show answer
Convert: 600 g = 0.60 kg. F = ma = 0.60 × 15 = 9.0 N. Check: 9.0 ÷ 0.60 = 15 m/s².
Where this leads next
Once F = ma is secure, explore what happens when two objects interact: momentum conservation in a simple closed model. If you are not sure when forces cancel, revisit balanced forces versus no forces.
Students who substitute the wrong force usually understand the equation but lack a routine for forming the resultant first. A teacher in online one-to-one Physics tuition can help build that routine around your own questions.