# Kinematics (SUVAT) calculator

> Kinematics calculator for the SUVAT equations: enter three of displacement, initial and final velocity, acceleration and time to get the other two.

Version interactive : https://www.calcopenly.com/fr/science/kinematics-calculator
Sujet : Calculatrices scientifiques

Motion with constant acceleration is described by five quantities: displacement s, initial velocity u, final velocity v, acceleration a and time t. Each of the five SUVAT equations leaves one of them out, so any three known values fix the other two. When time is unknown, s = ut + ½at² is a quadratic and can have two valid roots.

Physics students use it for braking, free-fall and launch problems. With the defaults, an object starting from rest and accelerating at 2 m/s² for 10 s reaches 20 m/s and covers 100 m. A ball thrown straight up at 20 m/s passes 15 m twice, at 0.99 s and 3.09 s, and both times are listed.

Acceleration must stay constant and the motion must lie along one line. Values are signed: choose a positive direction and give opposing quantities a minus sign, so free fall with up as positive uses a = −9.80665 m/s².

## Données

- **Find** (options : s and v (know u, a, t), v and t (know s, u, a), s and t (know u, v, a), s and a (know u, v, t), v and a (know s, u, t), a and t (know s, u, v), u and a (know s, v, t), u and t (know s, v, a), u and v (know s, a, t), s and u (know v, a, t))
- **Displacement s**: Signed: negative means behind the start point
- **Initial velocity u**
- **Final velocity v**
- **Acceleration a**: Use −9.80665 m/s² for free fall with up as positive
- **Time t**

## Résultats

- Final velocity v (m/s) — résultat principal
- Displacement s (m)
- Initial velocity u (m/s)
- Acceleration a (m/s²)
- Time t (s)
- Second time (other root) (s)
- Final velocity at the second time (m/s)
- Initial velocity for the second time (m/s)

## Formule

$$
v = u + at,\quad s = ut + \tfrac12at^2,\quad v^2 = u^2 + 2as,\quad s = \tfrac12(u+v)t,\quad s = vt - \tfrac12at^2
$$

## Exemples détaillés

### From rest at 3 m/s² for 8 s

- Find: s and v (know u, a, t)
- Initial velocity u: 0 m/s
- Acceleration a: 3 m/s²
- Time t: 8 s
- **Final velocity v: 24 m/s**
- **Displacement s: 96 m**
- Source de vérification : Python 3.8 decimal: v = 0 + 3·8 = 24; s = ½·3·8² = 96

### Ball thrown up at 20 m/s passes 15 m twice

- Find: v and t (know s, u, a)
- Displacement s: 15 m
- Initial velocity u: 20 m/s
- Acceleration a: -9.81 m/s²
- **Time t: 0.990719 s**
- **Second time (other root): 3.08675 s**
- **Final velocity v: 10.2811 m/s**
- **Final velocity at the second time: -10.2811 m/s**
- Source de vérification : Python 3.8 decimal: t = (20 ∓ √(400 − 2·9.81·15))/9.81 = 0.9907186…, 3.0867534…; v = 20 − 9.81t = ±10.2810505…

### Car braking from 10 m/s to rest in 50 m

- Find: a and t (know s, u, v)
- Displacement s: 50 m
- Initial velocity u: 10 m/s
- Final velocity v: 0 m/s
- **Time t: 10 s**
- **Acceleration a: -1 m/s²**
- Source de vérification : Python 3.8 decimal: t = 2s/(u+v) = 10; a = (0 − 100)/(2·50) = −1

### Zero acceleration (uniform motion)

- Find: v and t (know s, u, a)
- Displacement s: 100 m
- Initial velocity u: 5 m/s
- Acceleration a: 0 m/s²
- **Time t: 20 s**
- **Final velocity v: 5 m/s**
- Source de vérification : Python 3.8 decimal: t = s/u = 20 (linear case of ½at² + ut − s = 0)

### Launch speed to rise 20 m under gravity

- Find: u and t (know s, v, a)
- Displacement s: 20 m
- Final velocity v: 0 m/s
- Acceleration a: -9.81 m/s²
- **Initial velocity u: 19.8091 m/s**
- **Time t: 2.01928 s**
- Source de vérification : Python 3.8 decimal: u = √(0 + 2·9.81·20) = 19.80909…; t = u/9.81

### Speed from km/h and distance

- Find: s and t (know u, v, a)
- Initial velocity u: 36 km/h
- Final velocity v: 72 km/h
- Acceleration a: 2 m/s²
- **Time t: 5 s**
- **Displacement s: 75 m**
- Source de vérification : Python 3.8 decimal: 10 → 20 m/s at 2 m/s²: t = 5 s, s = (400 − 100)/4 = 75 m

## Questions

### What does SUVAT stand for?

SUVAT names the five quantities in the constant-acceleration equations: s for displacement, u for initial velocity, v for final velocity, a for acceleration and t for time. The equations are v = u + at, s = ut + ½at², v² = u² + 2as, s = ½(u + v)t and s = vt − ½at². Each omits a different variable, so you pick the one that leaves out the quantity you neither know nor need.

### When can you use the SUVAT equations?

Only when acceleration is constant and the motion is in a straight line. Free fall near the Earth's surface qualifies while air resistance is small; standard gravity is 9.80665 m/s², the value adopted by the 3rd General Conference on Weights and Measures in 1901. A skydiver nearing terminal velocity, or a car whose acceleration fades as it gains speed, needs calculus or a numerical model instead.

### How do you calculate stopping distance with SUVAT?

Set the final velocity to zero in v² = u² + 2as, which gives s = u² ÷ (2 × deceleration). A car braking from 10 m/s (36 km/h) at 1 m/s² stops in 50 m after 10 s. Doubling the starting speed quadruples the braking distance. The distance covered during the driver's reaction time comes on top and is uniform motion, s = ut.

### Why are there two answers for time?

When time is the unknown, s = ut + ½at² is a quadratic in t and can have two positive roots. A ball thrown upward at 20 m/s with a = −9.81 m/s² is 15 m above the start at 0.99 s on the way up and again at 3.09 s on the way down, moving at 10.28 m/s each time but in opposite directions. Negative roots are dropped because time starts at zero.

### What value of g should I use?

Use 9.80665 m/s², the defined standard gravity, unless the problem states another value; many textbooks round it to 9.81 or 9.8 m/s². Real sea-level gravity varies with latitude, from about 9.780 m/s² at the equator to 9.832 m/s² at the poles in the WGS 84 model. That spread changes answers by about 0.5%.

### Quelle est la précision de « Kinematics (SUVAT) calculator » ?

La précision dépend de vos données et des hypothèses de la méthode. Le calcul décimal utilise 50 chiffres significatifs, mais les estimations, méthodes numériques et données sources peuvent être moins précises ; l’arrondi affiché ne supprime pas ces limites. Exemples résolus vérifiés à partir de sources indépendantes : 6. Par exemple, « From rest at 3 m/s² for 8 s » est vérifié à l’aide de Python 3.8 decimal: v = 0 + 3·8 = 24; s = ½·3·8² = 96.

### D’où vient cette méthode ?

OpenStax University Physics Volume 1, §3.4 Motion with constant acceleration; HyperPhysics — Motion equations for constant acceleration.

## Sources

- [OpenStax University Physics Volume 1, §3.4 Motion with constant acceleration](https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration)
- [HyperPhysics — Motion equations for constant acceleration](http://hyperphysics.phy-astr.gsu.edu/hbase/mot.html)