Average Force Formula: F = Δp / Δt with Solved Examples

The average force formula is $F_{\text{avg}} = \frac{\Delta p}{\Delta t} = \frac{m(v - u)}{t}$ : divide the change in momentum by the time it took. A 0.45 kg football kicked from rest to 20 m/s in 0.010 s feels an average force of 900 N from the boot.

Real forces in kicks, crashes and catches rise and fall within milliseconds. The average force is the steady force that would give the same change in momentum over the same time. It is the practical form of Newton's second law, the central result of the laws of motion, for events too fast to follow in detail.

Quick facts

Time average
$F_{\text{avg}} = \frac{\Delta p}{\Delta t}$
Distance average
$F_{\text{avg}} = \frac{\Delta K}{d}$
SI unit
newton (N)
Dimensional formula
[M¹ L¹ T⁻²]
From a graph
Area under F against t, divided by the time
Type
Vector, along the change in momentum

Average force formula explained

Symbols in F_avg = m(v − u)/t
Symbol	Meaning	Unit
$F_{\text{avg}}$	Average net force	N
$m$	Mass being pushed	kg
$u$	Velocity when the push begins	m/s
$v$	Velocity when the push ends	m/s
$t$	Duration of the push	s

The numerator is the change in momentum, which also equals the impulse. So another way to write it is $F_{\text{avg}} = \frac{J}{\Delta t}$ . For a constant force the average is simply that force, and the formula reduces to $F = ma$ .

Worked examples

Example 1: kicking a football

A 0.45 kg ball at rest is kicked to 20 m/s. Contact lasts 0.010 s. Find the average force.

$F_{\text{avg}} = \frac{0.45 \times (20 - 0)}{0.010} = \frac{9.0}{0.010} = 900\ \text{N}$

Example 2: crash barrier, with km/h

A 1200 kg car moving at 54 km/h hits a barrier and stops in 0.15 s. Find the average force on the car.

Convert: 54 ÷ 3.6 = 15 m/s.

$F_{\text{avg}} = \frac{1200 \times (0 - 15)}{0.15} = -1.2 \times 10^5\ \text{N}$

The barrier pushes back with about 120 kN. If crumple zones stretched the stopping time to 0.30 s, the force would halve.

Example 3: a force that changes in steps

A cart is pushed with 10 N for 2.0 s and then 4.0 N for 3.0 s, in the same direction. Find the average force over the 5.0 s.

Total impulse: $10 \times 2.0 + 4.0 \times 3.0 = 20 + 12 = 32\ \text{N s}$ .

$F_{\text{avg}} = \frac{32}{5.0} = 6.4\ \text{N}$

Averaging 10 N and 4 N directly gives 7 N, which is wrong because the forces act for different times.

Time average and distance average

There are two averages, and questions do not always say which. The time average uses momentum, $F = \frac{\Delta p}{\Delta t}$ . The distance average uses energy, from the work energy theorem: $F = \frac{\Delta K}{d}$ .

Example 4: a pellet stopping in wood

A 10 g pellet travelling at 300 m/s comes to rest 6.0 cm inside a wooden block. Find the average resisting force.

$K = 0.5 \times 0.010 \times 300^2 = 450\ \text{J}$ , and d = 0.060 m.

$F = \frac{450}{0.060} = 7.5 \times 10^3\ \text{N}$

Use time when the time is given and distance when the distance is given. For a steady force both give the same answer.

Reading average force from a graph

Plot F up the side and t along the bottom; the area enclosed is the impulse. Divide that area by the total time to get the average force. Geometrically, this is the height of the rectangle with the same base and the same area as the real curve. A triangular pulse peaking at 600 N has an average force of 300 N over its duration. The change in momentum formula page has more on these graphs.

Errors with average force

Averaging the forces without weighting each by its time.
Leaving milliseconds unconverted.
Losing the minus sign in a stopping problem, then quoting the wrong direction.
Mixing the time average and the distance average in one calculation.

Safety devices never change Δp in a crash: the car still goes from full speed to zero. They only lengthen Δt, which cuts the average force.

For a single constant force, the force formula and acceleration from force pages are simpler starting points.

Frequently asked questions

What is the average force formula?

Average force equals change in momentum divided by time, F_avg = m(v − u) / t. It is the steady force that would produce the same change in the same time.

How do you find average force from distance?

Divide the change in kinetic energy by the distance over which the force acts: F = ΔK / d. This comes from the work energy theorem.

What is the difference between average force and impulse?

Impulse is F_avg × Δt and is quoted in N s. Average force is impulse per second of contact, quoted in N.

Why does a longer impact time reduce the force?

The change in momentum is fixed by the speeds before and after. Spreading it over more time means less force each instant, since F = Δp / Δt.