# 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.

Interactive version: https://www.calcopenly.com/science/kinematics-calculator
Subject: Science calculators

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.

## Inputs

- **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**

## Results

- Final velocity v (m/s) — main result
- 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)

## Formula

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

## Worked examples

### 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**
- Checked against: 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**
- Checked against: 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**
- Checked against: 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**
- Checked against: 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**
- Checked against: 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**
- Checked against: 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%.

### How accurate is the kinematics (SUVAT) calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 6 worked examples whose answers come from independent sources; for example, “From rest at 3 m/s² for 8 s” is checked against Python 3.8 decimal: v = 0 + 3·8 = 24; s = ½·3·8² = 96.

### Where does the method come from?

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)