Kinematic Equations Calculator

Compute velocity, displacement, acceleration or time with the four SUVAT equations for uniformly accelerated motion. Four modes with stepwise solution.

Calculation parameters
Scegli quale variabile mancante vuoi calcolare.
Velocità lungo l'asse del moto all'istante t=0. In SI si misura in metri al secondo.
Velocità finale al termine dell'intervallo. Lascia vuoto nei modi in cui v è l'incognita.
Accelerazione costante lungo l'asse del moto. Per caduta libera si usa g = 9,80665 m/s².
Intervallo temporale tra l'istante iniziale e quello finale.
Distanza percorsa lungo l'asse del moto. È una grandezza vettoriale, positiva nella direzione scelta come positiva.
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The SUVAT kinematic equations

The kinematic equations, also known as SUVAT equations from the initials of the five quantities they relate (s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time), describe one-dimensional motion with constant acceleration. They are the backbone of introductory mechanics: free fall near the Earth's surface, a car braking at constant deceleration, an object launched vertically upward, a block sliding on a frictionless incline — all of these are modelled exactly by SUVAT as long as the acceleration does not change during the time interval of interest.

The four equations

Under constant acceleration, four algebraically equivalent relations connect any four of the five SUVAT variables, omitting the fifth:

  1. v = u + a·t (missing s) — definition of average acceleration.
  2. s = u·t + ½·a·t² (missing v) — from double integration of the acceleration.
  3. v² = u² + 2·a·s (missing t) — work-energy theorem in kinematic form.
  4. s = ½·(u + v)·t (missing a) — displacement as average velocity times time.

A fifth form, s = v·t − ½·a·t² (missing u), is equivalent to (2) and rarely needed.

How the calculator works

The tool offers four modes, one for each combination of three known inputs, and returns the remaining two quantities together with the equation used and the algebraic steps:

  • Given u, a, t → finds v via (1) and s via (2).
  • Given u, v, a → finds t from (1) and s from (3).
  • Given u, v, t → finds a from (1) and s from (4).
  • Given u, a, s → finds v from (3) then t from (1). If the discriminant u²+2as is negative, the inputs are physically inconsistent.

Sign conventions

All vectors — u, v, a, s — are projections onto a pre-chosen oriented axis. A positive value means the quantity points along the axis; a negative value means it points the opposite way. For free fall with an axis pointing up, gravitational acceleration is a = −9.80665 m/s²; if the axis points down, a = +9.80665 m/s². Decide the positive direction before entering numbers.

Limits of applicability

SUVAT works only when acceleration is constant. It does not apply to simple harmonic motion (where a depends on position), to fall with velocity-dependent air drag, to gravitational orbits (a varies with distance), or to any motion driven by a time-varying force. In those cases one integrates Newton's equations directly, often numerically.

Units

The calculator works in SI units: metres (m) for displacement, seconds (s) for time, metres per second (m/s) for velocities, metres per second squared (m/s²) for acceleration. Convert km/h to m/s by dividing by 3.6, mph to m/s by multiplying by 0.44704.

Worked examples

Braking car. A car travelling at 108 km/h (30 m/s) decelerates at −6 m/s² to a stop. Using mode u, v, a → t, s: t = (0−30)/(−6) = 5.000 s; s = (0²−30²)/(2·(−6)) = 75 m.

Free fall. A stone dropped from rest for 3 s with g = 9.80665 m/s². Mode u, a, t → v, s with u = 0, a = 9.80665, t = 3: v = 29.42 m/s; s = 44.13 m.

Common mistakes to avoid

The most frequent error is applying the SUVAT equations to motion in which acceleration is not constant: they hold only for uniformly accelerated motion. A second mistake is confusing signs: if you choose upward as positive, gravity must be entered as −9.81 m/s². Be careful not to mix units (km/h with m/s²), and remember that displacement s is a vector, not the distance travelled: in motion that reverses direction the two values differ. Finally, in the v² = u² + 2as mode the square root gives two solutions: you must pick the one that is physically meaningful for the direction of motion.

Practical applications of uniformly accelerated motion

The kinematic equations are an everyday tool of engineering and applied physics. They are used to compute vehicle braking distances (essential for road safety), to size aircraft take-off and landing runways, to study free fall and projectile motion in ballistics, and to design elevators, hoists and transport systems with comfortable accelerations. In sport they describe a sprinter's start or the flight of a ball; in astronautics the lift-off of rockets during the constant-thrust phase. They are also the didactic foundation on which all later kinematics and dynamics are built.

Updated July 2026

Frequently Asked Questions

What is the difference between average and instantaneous velocity?

Average velocity is the total displacement divided by the total time, while instantaneous velocity is the velocity at a precise instant. In uniformly accelerated motion the average velocity equals the arithmetic mean of the initial and final velocities, (u+v)/2, and corresponds to the instantaneous velocity at the midpoint in time. They are equal only when the acceleration is zero, that is in uniform straight-line motion.

Do the equations also work for decelerated motion?

Yes, you just enter the acceleration with a negative sign relative to the direction of motion. Deceleration is simply an acceleration opposite to the velocity: a car braking from 30 m/s at a = −6 m/s² stops in 5 seconds. The same four equations describe both acceleration and braking; the only care needed is consistency of signs with the chosen reference frame.

What value of g should I use for free fall?

The standard value of gravitational acceleration is g = 9.81 m/s² (more precisely 9.80665 m/s²). You enter it as the acceleration, with a positive sign if you choose downward as positive, negative if upward. The value varies slightly with latitude and altitude, from about 9.78 at the equator to 9.83 at the poles, but 9.81 is an excellent approximation for most calculations.

Can I enter units other than SI?

The calculator works in SI units (metres, seconds, m/s, m/s²), so it is best to convert your data first. Remember that 1 km/h = 0.2778 m/s (divide by 3.6) and 1 mph ≈ 0.447 m/s. If you enter quantities in units consistent with each other the result is correct, but mixing km/h with m/s² produces errors: good practice is to always convert everything to SI units before calculating.

What does a negative discriminant mean in mode u, a, s?

In the formula v² = u² + 2as, if the quantity u² + 2as turns out negative there is no real velocity: the square root of a negative number is undefined in the real numbers. Physically it means the imposed conditions are impossible, for example asking a body to reach a displacement that its initial velocity and deceleration do not allow it to travel. In that case the motion stops earlier.

Do these equations work in 2D or 3D?

The SUVAT equations are one-dimensional, but two- or three-dimensional motion is handled by decomposing it along the axes. In projectile motion, for instance, the equations are applied separately to the horizontal axis (zero acceleration, uniform motion) and to the vertical axis (gravitational acceleration). Each component follows its own SUVAT equations, and they are combined at the end to obtain the trajectory and resultant velocity.

Why four equations and not three?

There are five quantities involved (s, u, v, a, t) and each equation relates four of them, omitting one. The four fundamental equations correspond to omitting each variable in turn (apart from one redundant combination): so whatever the unknown and whichever three are known, there is always a suitable equation. It is this completeness that makes the SUVAT set such a versatile tool.

What are kinematic equations used for in real life?

They are used wherever constant-acceleration motion is studied: in computing braking distances for road safety, sizing airport runways, in ballistics and shooting, in the design of lifts and fairground rides, in the analysis of jumps and sprints in sport, and in the thrust phase of rockets. They are also the didactic foundation of all the physics of motion, from which dynamics and energy are developed.

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