Kinematics (SUVAT), Explained
After reading this you can solve any constant-acceleration motion problem from three known quantities, check your answer against a chart, and know exactly when the equations stop applying.
What SUVAT solves
Drop a ball off a 44 m cliff. How long until it hits the ground, and how fast is it moving when it lands? You do not need calculus for this. As long as the acceleration stays constant (here, gravity at about 9.81 m/s²), five algebraic equations connect everything you might want to know.
Those five quantities are the letters in SUVAT:
- s
- displacement, the straight-line distance from start to finish, in metres. Not the same as total path length.
- u
- initial velocity, in m/s.
- v
- final velocity, in m/s.
- a
- acceleration, in m/s², assumed constant.
- t
- elapsed time, in seconds.
Here is the key fact. These five variables are not independent. Given any three of them, the other two are fixed. That is why the calculator asks for exactly three inputs. The falling-ball problem gives you u = 0, a = 9.81, and s = 44, which is three, so t and v follow.
When to use it, and when not to
Use SUVAT when acceleration is constant in both magnitude and direction along a straight line. Free fall near the Earth's surface qualifies, because gravity varies by less than 0.1% over the first few kilometres of height. A car braking with a fixed pedal force is a reasonable approximation. A trolley on a tilted track qualifies if friction is steady.
Do not use it when acceleration changes during the motion. Air resistance on a fast object grows with speed, so a skydiver's acceleration falls from 9.81 m/s² toward zero at terminal velocity. SUVAT gives wrong answers there. A rocket burning fuel loses mass, so its acceleration climbs even at constant thrust. Circular motion has an acceleration that constantly changes direction, which is why it gets its own Circular Motion Calculator. Springs and pendulums have acceleration that varies with position, handled by the Pendulum & Spring Period Calculator.
The single most common misuse is applying one SUVAT equation across a phase change. If a car accelerates for 4 s then brakes for 3 s, you cannot span both phases with one equation. Split the problem: solve the acceleration phase, then use its final velocity as the initial velocity of the braking phase.
The five equations and where they come from
Every SUVAT equation follows from two definitions. Acceleration is the rate of change of velocity, and for constant acceleration the average velocity is the mean of start and end values.
Start with the definition of constant acceleration:
Final velocity v equals the starting velocity u plus the gain at. This is the one equation with no s in it.
Because acceleration is constant, the velocity climbs in a straight line, so the average velocity is (u+v)/2. Distance is average velocity times time:
This is the equation with no a in it. Substitute v = u + at into it to eliminate v, and you get the equation with no v:
The first term ut is how far you would travel at the starting speed. The second term \frac{1}{2}at^2 is the extra distance the acceleration adds. Instead substitute u = v - at to get the equation with no u:
Finally, eliminate t altogether by combining the first two. This is the equation with no t, the one you reach for when time is neither known nor wanted:
Each equation leaves out exactly one of the five variables. Pick the equation missing the variable you neither know nor need.
Worked example: the demo data
An object released from rest and falling for 3 seconds
The demo button loads u = 0 m/s, a = 9.81 m/s², and t = 3 s. That is three known values, so the calculator returns v and s.
- Final velocity uses v = u + at. So v = 0 + 9.81 \times 3 = 29.43 m/s.
- Displacement uses s = ut + \frac{1}{2}at^2. The first term is 0 \times 3 = 0. The second is \frac{1}{2} \times 9.81 \times 3^2 = 0.5 \times 9.81 \times 9 = 44.145 m.
- Check with the no-time equation: v^2 = u^2 + 2as = 0 + 2 \times 9.81 \times 44.145 = 866.2, and \sqrt{866.2} = 29.43 m/s. It matches, so the answers are consistent.
So the ball reaches 29.43 m/s and falls 44.15 m in 3 seconds. Notice that the cliff in the opening hook was 44 m, which is why the ball took just under 3 seconds to land.
Try changing the acceleration
The shape of the distance curve depends entirely on a. On the Moon gravity is about 1.62 m/s², so the same 3-second fall covers only \frac{1}{2} \times 1.62 \times 9 = 7.29 m. On Jupiter, near 24.8 m/s², it covers about 112 m. The widget below lets you feel that difference.
Reading and interpreting results
Watch the signs. SUVAT works on a signed axis, so you must choose a positive direction and stick with it. If you call downward positive for the falling ball, then a is +9.81. If you call upward positive, then a is -9.81 and a ball thrown up at +20 m/s has a negative displacement once it passes back below the launch point.
A negative t from the quadratic equations usually means you picked a solution that happens before your clock starts. The v^2 = u^2 + 2as equation gives two roots for v (one positive, one negative) because it only knows the square. Pick the sign that matches the physical direction of travel.
When you solve s = ut + \frac{1}{2}at^2 for time, you are solving a quadratic in t. Two positive roots can both be real. A ball thrown upward passes a given height once on the way up and once on the way down, so both times are correct answers to the same equation.
| Equation | Missing variable | Use when you do not have or want |
|---|---|---|
| v = u + at | s | displacement |
| s = \frac{(u+v)}{2}t | a | acceleration |
| s = ut + \frac{1}{2}at^2 | v | final velocity |
| s = vt - \frac{1}{2}at^2 | u | initial velocity |
| v^2 = u^2 + 2as | t | time |
Common mistakes
Mixing units is the usual trap. If you are given a speed in km/h and a time in seconds, convert first. A car at 72 km/h is 72 \times 1000 / 3600 = 20 m/s. Feed 72 into a SUVAT equation expecting m/s and your braking distance is off by a factor of 3.6.
Confusing displacement with distance is the second trap. A ball thrown straight up 10 m and caught back at the start has travelled 20 m of path but has a displacement of 0. SUVAT tracks displacement, so s = 0 for the full up-and-down trip.
The third trap is assuming an object "starts from rest" when it does not. Read the problem. A car braking to a stop has v = 0 at the end, not u = 0 at the start. Getting these two backwards flips your acceleration sign.
Once you know the acceleration, the force behind it comes from the Force Calculator. If the motion involves an inclined surface, the Friction & Inclined Plane Calculator gives you the acceleration to feed back into SUVAT.
Related tools
SUVAT gives you velocities and displacements, which are the raw material for other mechanics. Once you have the final velocity, the Kinetic & Potential Energy Calculator turns it into kinetic energy, and the Momentum & Collision Calculator turns it into momentum for a collision. If you know the force and the distance instead, the Work & Power Calculator connects the two. For rotational problems there is the Torque Calculator, and for fluids the Pressure & Buoyancy Calculator and the Density Calculator.
Frequently asked questions
Why do I need exactly three values?
The five variables obey two independent relationships, so two of them are always determined by the other three. Give fewer than three and the problem has infinitely many solutions. Give an inconsistent set of four or five and you may get a contradiction.
What value should I use for gravity?
Near the Earth's surface, 9.81 m/s² is standard. Many textbooks round to 9.8 or even 10 for mental arithmetic. The difference between 9.81 and 10 is about 1.9%, so a 44 m answer becomes about 43.2 m, close but not identical.
Does SUVAT work for objects thrown at an angle?
Yes, but you must split the motion into horizontal and vertical components and apply SUVAT to each separately. Horizontal acceleration is 0 (ignoring air), so horizontal velocity stays constant. Vertical acceleration is -9.81 m/s². The two share the same time t, which links them.
Can acceleration be negative?
Yes. A negative acceleration simply means the acceleration points in your chosen negative direction. If an object moves in the positive direction and its acceleration is negative, it is slowing down. If both are negative, it is speeding up while moving backward.
Why does my quadratic give two times?
The displacement equation s = ut + \frac{1}{2}at^2 is second order in t. For a projectile that rises and falls, both roots are physically real: the object passes a given height twice. Discard a negative root only if it lies before your start time.
Related tools
Force Calculator (F = ma)
Solve Newton's second law for force, mass or acceleration.
Kinetic & Potential Energy Calculator
KE = ½mv², PE = mgh and total mechanical energy.
Projectile Motion Calculator
Range, height, flight time and the trajectory curve.
Circular Motion Calculator
Centripetal acceleration, force and angular speed.
Density Calculator (ρ = m/V)
Solve density, mass or volume from the other two.
Free Fall Calculator
Fall time and impact speed from a height, or vice versa.