Acceleration: Definition, Formula a = Δv/Δt, and Examples

Acceleration is the rate at which velocity changes. Because velocity is a vector, any change in either the speed or the direction of motion constitutes acceleration. Every time you brake in a car, ride a carousel, throw a ball, or fire a rocket, acceleration is at work. It is the physical quantity that links force to motion through Newton's second law (F = ma) — arguably the most important equation in classical mechanics.

Acceleration — Definition

Acceleration is the rate of change of velocity with respect to time. Formula: a = Δv/Δt. It is a vector quantity with both magnitude and direction, measured in metres per second squared (m/s²). Positive acceleration means velocity is increasing in the positive direction; negative acceleration (deceleration) means velocity is decreasing or directed opposite to the chosen positive direction.

The Acceleration Formula

a = Δv / Δt = (v_final − v_initial) / t

The unit m/s² is intuitive: acceleration of 1 m/s² means velocity changes by 1 m/s every second. A car accelerating from rest at 3 m/s² will reach 3 m/s after 1 s, 6 m/s after 2 s, 9 m/s after 3 s, and so on.

Types of Acceleration

Uniform (constant) acceleration

Velocity changes by the same amount every second. Free fall near Earth's surface is the canonical example: in the absence of air resistance, every object falls with constant acceleration g = 9.8 m/s² downward. The SUVAT equations describe all constant-acceleration motion.

Non-uniform acceleration

Most real-world cases involve varying acceleration — a car in traffic, a rocket burning fuel (mass decreasing so acceleration increases for the same thrust). For non-uniform acceleration, instantaneous acceleration is:

a = dv/dt = d²s/dt²

Centripetal acceleration

For an object in circular motion with radius r at speed v:

a_c = v²/r (directed toward the centre)

This changes the direction of velocity without changing its magnitude. It keeps planets in orbit, cars on curved roads, and electrons in cyclotrons. By Newton's second law, centripetal force = mv²/r, directed toward centre.

Acceleration and Newton's Second Law

F_net = ma → a = F_net / m

Acceleration is directly proportional to net force and inversely proportional to mass. Double the force → double the acceleration. Double the mass → halve the acceleration. A lorry accelerates more slowly than a car under the same engine force because it has greater mass.

Scenario	Net Force	Acceleration
Free fall (no air resistance)	mg downward	9.8 m/s² downward
Constant velocity	0 N	0 m/s²
Braking car	Friction backward	Negative (deceleration)
Circular orbit	Gravity toward centre	v²/r toward centre

Acceleration Due to Gravity: g = 9.8 m/s²

g ≈ 9.8 m/s² (downward, near Earth's surface)

Any object in free fall — regardless of mass — accelerates at g = 9.8 m/s². A feather and a hammer fall identically in vacuum, as Apollo 15 astronaut David Scott famously demonstrated on the Moon in 1971. This mass-independence was Galileo's great discovery and the seed of Einstein's general relativity.

g varies slightly: 9.832 m/s² at the poles, 9.780 m/s² at the equator, ~9.77 m/s² atop Everest. For most problems, g = 9.8 m/s² or g = 10 m/s² (approximate) is used.

Worked Examples

Example 1: Car acceleration

0 to 30 m/s in 10 seconds: a = (30 − 0) / 10 = 3 m/s²

Example 2: Braking (negative acceleration)

25 m/s to 0 in 5 seconds: a = (0 − 25) / 5 = −5 m/s²

Example 3: Newton's second law

1,200 kg car, 3,600 N net force: a = 3600 / 1200 = 3 m/s²

Example 4: Centripetal acceleration

Roundabout of radius 40 m at 15 m/s: a_c = 15² / 40 = 5.625 m/s² toward centre

SUVAT Equations for Constant Acceleration

Four equations link the five kinematic variables s (displacement), u (initial velocity), v (final velocity), a (acceleration), t (time):

v = u + at

s = ut + ½at²

v² = u² + 2as

s = ½(u + v)t

Know any three variables → find the other two. These equations underpin all of projectile motion analysis and are among the most-used equations in introductory physics.

Frequently Asked Questions

What is acceleration in physics?

Acceleration is the rate of change of velocity with respect to time: a = Δv/Δt. It is a vector — it has magnitude and direction. Any change in speed or direction is acceleration. Its SI unit is m/s².

What is the formula for acceleration?

Average acceleration: a = Δv/Δt = (v_f − v_i)/t. Instantaneous: a = dv/dt. Centripetal: a_c = v²/r. All measured in m/s².

Can acceleration be negative?

Yes. Negative acceleration means the acceleration vector points opposite to the velocity vector, causing the object to slow down. It is not "less acceleration" — it is acceleration in a specific direction. Braking produces negative acceleration when forward is the positive direction.

What is the acceleration due to gravity?

Near Earth's surface, g ≈ 9.8 m/s² directed downward. All objects in free fall experience this acceleration regardless of mass. g varies slightly with altitude and latitude: ~9.78 m/s² at the equator, ~9.83 m/s² at the poles.

Can an object accelerate without changing speed?

Yes. Any change in direction is a change in velocity — and therefore an acceleration — even if speed is constant. An object in circular motion is continuously accelerating toward the centre, while its speed remains unchanged. This centripetal acceleration requires a centripetal force.

Is acceleration a vector or scalar?

Acceleration is a vector — it has both magnitude and direction. Its direction is the same as the net force on the object (from Newton's second law). Centripetal acceleration points toward the centre of the circle; gravitational acceleration points downward toward Earth's centre.