Acceleration Calculator
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Centripetal acceleration always points towards the centre of the circular path.
Acceleration Reference Values
Understanding Acceleration in Physics
Acceleration is one of the core concepts in Newtonian mechanics, appearing in virtually every dynamics problem in GCSE and A-Level Physics. It describes how quickly velocity changes, and is central to understanding forces, motion, and circular dynamics.
Defining Acceleration
Acceleration (a) is the rate of change of velocity with time: a = Δv / Δt = (v − u) / t. Since velocity is a vector, so is acceleration. An object accelerates whenever its velocity changes — this includes speeding up, slowing down (deceleration), or changing direction (even at constant speed, as in circular motion). The SI unit is m/s² (metres per second squared).
Newton's Three Laws of Motion
First Law (Inertia): An object remains at rest or moves at constant velocity unless acted upon by a resultant (net) force. This means zero net force = zero acceleration.
Second Law (F = ma): The resultant force on an object equals its mass times its acceleration: F = ma. This allows calculation of any one of F, m, or a when the other two are known. A 1 N force produces 1 m/s² acceleration in a 1 kg mass.
Third Law (Equal and Opposite): Every action has an equal and opposite reaction. Forces act in pairs on different objects — important for understanding momentum and collisions.
Kinematic Acceleration: a = (v-u)/t
This is the most direct definition. Given: initial velocity u, final velocity v, and time t, the (uniform) acceleration is a = (v−u)/t. Rearrangements:
- v = u + at — Final velocity after accelerating from u for time t
- u = v − at — Initial velocity if final state is known
- t = (v−u) / a — Time to achieve a velocity change
A negative result for a indicates deceleration (velocity is decreasing in the positive direction).
Newton's Second Law: F = ma
The net force (resultant force) is the vector sum of all forces acting. For a car: Fnet = Fengine − Ffriction − Fdrag. Rearrangements: a = F/m (large forces or small masses produce large accelerations) and m = F/a (find mass from force and acceleration). The weight of an object on Earth is W = mg = m × 9.81 N.
Centripetal Acceleration: a = v²/r
For an object moving in a circle of radius r at speed v, the centripetal acceleration points inward (toward the centre): a = v²/r. The centripetal force required to maintain this motion is F = mv²/r. Common examples:
- Car on a roundabout: centripetal force provided by friction
- Satellite in orbit: centripetal force provided by gravity
- Electron in atom: centripetal force provided by electrostatic attraction
Alternative formula using angular velocity ω (rad/s): a = ω²r.
Worked Examples
Example 1 (Kinematic): A train accelerates from 10 m/s to 30 m/s in 20 s.
a = (30 − 10) / 20 = 1 m/s²
Example 2 (Newton): A 800 kg car has a net force of 2,400 N applied.
a = F / m = 2400 / 800 = 3 m/s²
Example 3 (Centripetal): A ball on a 2 m string moves at 4 m/s.
a = v² / r = 16 / 2 = 8 m/s² (directed toward the centre)
Graphs of Motion
On a velocity-time (v-t) graph: the gradient = acceleration; area under graph = displacement. On a displacement-time (s-t) graph: gradient = velocity; a curved line indicates changing velocity (acceleration). These graphical methods often appear in UK Physics exam mark schemes as alternative solution routes.
Real-World Applications of Acceleration
Understanding acceleration is essential in many real-world contexts: vehicle safety design (crumple zones and airbags reduce acceleration on impact), sports science (measuring sprinting acceleration), aerospace engineering (calculating rocket thrust requirements), and transportation planning (stopping distances depend on deceleration). In the UK, stopping distance calculations for driving theory tests involve constant deceleration problems directly related to a = (v−u)/t.
Momentum and Impulse
Related to F = ma: momentum p = mv (kg·m/s). The impulse-momentum theorem states FΔt = Δp = mΔv. Impulse (N·s) equals the change in momentum. This is why airbags work: by increasing collision time, they reduce the force (even though the impulse is the same). Conservation of momentum states the total momentum of a closed system is constant, regardless of internal forces.