Force Formula: F = ma, SI Unit, Examples and Calculator

The force formula is $F = ma$ : the net force on a body equals its mass multiplied by the acceleration that force produces. Push a 2 kg cart so that it speeds up by 3 m/s every second and the unbalanced push on it must be 6 newtons.

This is Newton's second law written for a body whose mass stays fixed. It is the first equation most students learn in the laws of motion chapter, from Class 9 force and Class 11 dynamics in India to the dynamics unit of AP Physics 1. Use the calculator below to solve for any one of the three quantities once you know the other two.

Quick facts

Formula
$F = ma$
SI unit
newton (N), where 1 N = 1 kg m/s²
Dimensional formula
[M¹ L¹ T⁻²]
Type
Vector, it points the same way as the acceleration
General form
$F = \frac{\Delta p}{\Delta t}$ , rate of change of momentum
CGS unit
dyne, with 1 N = 10⁵ dyne

The force formula F = ma, symbol by symbol

Symbols in F = ma
Symbol	What it stands for	SI unit
$F$	The net force, the vector sum of every force acting on the body	N
$m$	Mass, the amount of matter and the body's resistance to a change in motion	kg
$a$	Acceleration, the rate at which velocity changes	m/s²

The F in this equation is not any single push or pull. It is what is left after all the forces are added with their directions. A crate pushed with 50 N while friction holds it back with 30 N has a net force of 20 N, and only that 20 N goes into $F = ma$ . The net force formula page shows how to add forces that act along different lines.

One newton is the force that gives a 1 kg mass an acceleration of 1 m/s². So $1\ \text{N} = 1\ \text{kg m/s}^2$ . An apple held in your hand weighs roughly one newton, which helps you sense whether an answer is sensible.

Force calculation formula: five steps to find force

Draw the body alone. Sketch it as a dot or box and mark every force on it with an arrow.
Choose a positive direction. Usually the direction of motion. Forces the other way get a minus sign.
Convert to SI. Speeds in km/h become m/s once you divide by 3.6; grams become kilograms once you divide by 1000.
Find the acceleration if it is not given. Use $a = \frac{v - u}{t}$ or another equation of motion.
Multiply. $F = ma$ , with the sign kept. A negative force acts against your positive direction.

If you need the acceleration rather than the force, the a = F/m page works the rearranged form $a = \frac{F}{m}$ in detail.

Solved examples on the formula of force

Example 1: a shopping trolley

A trolley of mass 12 kg speeds up at 1.5 m/s². What net force acts on it?

Given: m = 12 kg, a = 1.5 m/s².

$F = ma = 12 \times 1.5 = 18\ \text{N}$

The net force is 18 N in the direction of the acceleration.

Example 2: braking a car, with km/h

A car of mass 1200 kg travelling at 90 km/h stops in 5.0 s. Find the braking force.

First change the speed: 90 ÷ 3.6 gives 25 m/s. Now u = 25 m/s, v = 0 and t = 5.0 s.

$a = \frac{0 - 25}{5.0} = -5.0\ \text{m/s}^2$

$F = 1200 \times (-5.0) = -6000\ \text{N} = -6.0\ \text{kN}$

The minus sign shows the force acts backward, against the motion. The braking force has size 6.0 kN.

Example 3: catching a cricket ball

A 0.16 kg ball moving at 30 m/s is stopped by a fielder's hands in 0.050 s. What average force do the hands exert?

Here the momentum route is quicker: $F = \frac{\Delta p}{\Delta t} = \frac{0.16 \times (0 - 30)}{0.050}$ .

$F = \frac{-4.8}{0.050} = -96\ \text{N}$

The hands push back on the ball with about 96 N. Drawing the hands back makes the time longer and the force smaller.

Other formulas for force you will meet

Many chapters have their own formula of force. Each one still obeys the second law, so its unit is always the newton.

Common force formulas by chapter
Force	Formula	Page
Momentum changing over time	$F = \frac{\Delta p}{\Delta t}$	Change in momentum
Weight	$W = mg$	Weight
Friction	$f = \mu N$	Friction force
Spring	$F = -kx$	Spring force
Pull between two masses	$F = \frac{GMm}{r^2}$	Newton's gravitation
Circular motion	$F = \frac{mv^2}{r}$	Centripetal force
Electric charge	$F = \frac{k q_1 q_2}{r^2}$	Electrostatic force
From pressure	$F = PA$	Pressure

The momentum form $F = \frac{dp}{dt}$ is the more general statement of the law. It reduces to $F = ma$ only when the mass does not change. For a rocket burning fuel, or sand falling onto a moving belt, you need the momentum form.

Units and dimensions of force

Since force is mass times acceleration, its SI unit is kg × m/s², named the newton. In the CGS system the unit is the dyne, the force that gives 1 g an acceleration of 1 cm/s². Converting: $1\ \text{N} = 1000\ \text{g} \times 100\ \text{cm/s}^2 = 10^5\ \text{dyne}$ . Older Indian questions also use the kilogram force, the weight of 1 kg, so 1 kgf = 9.8 N.

In base dimensions, mass is [M] and acceleration is [L T⁻²], giving [M¹ L¹ T⁻²]. The page on the dimensional formula of force sets out every line of that working. Tension, thrust, weight and friction share these dimensions because each is a force.

Mistakes students make with F = ma

Using one force instead of the net force. Subtract friction or add weight first, then apply the formula.
Mass in grams. 250 g is 0.25 kg. Forgetting this makes the answer 1000 times too big.
Confusing mass and weight. Mass is in kg, weight is a force in N. A 60 kg student has a weight close to 590 N.
Thinking a moving body needs a force. A puck sliding at steady speed on smooth ice has zero net force. Force changes motion; it is not needed to keep motion going.
Ignoring direction. Force is a vector. A negative answer is not an error; it tells you which way the force points.

Sense check: a person pushing a heavy box manages a few hundred newtons at most. An answer in meganewtons for an everyday push almost always hides a unit slip.

Frequently asked questions

What is the formula of force in physics?

Force equals mass times acceleration, F = ma. F stands for the unbalanced force (N), m for mass (kg) and a for acceleration (m/s²).

Which SI unit is force measured in?

The SI unit is the newton, symbol N. One newton is 1 kg m/s², the force that accelerates a 1 kg mass at 1 m/s².

Is force a scalar or a vector?

Force is a vector quantity. It has a size in newtons and a direction, and forces must be added as vectors to find the net force.

What is the formula to find force without acceleration?

Use the change in momentum over time, F = (mv − mu) / t. You can also use a specific law such as W = mg for weight or f = μN for friction.

What is the difference between force and pressure?

Force is a push or pull measured in newtons. Pressure is force spread over an area, P = F / A, measured in pascals.