SUVAT is the shorthand British physics courses use for the five equations of motion under constant acceleration, named after the five symbols they connect: s for displacement, u for initial velocity, v for final velocity, a for acceleration and t for time. Each equation leaves out exactly one of the five, which is the whole point of having five of them. Knowing three quantities is always enough to find the other two, and choosing the right equation is simply a matter of picking the one that omits the quantity you neither know nor want.
This SUVAT calculator from Arb Digital does the selection for you and then tells you which equation it used, because the equation name is usually the part you are being marked on. It also does something most calculators of this type skip: when the three values you supplied cannot determine the other two, or contradict each other outright, it says so plainly instead of returning a plausible-looking number that happens to be wrong.
What This SUVAT Calculator Does
You type any three of the five quantities and leave the other two blank. The tool works out which of the ten possible combinations you have given it, applies the correct rearrangement, and reports both missing values in the result hero. The four-item grid then shows the complete solved set — s, v, a and t — so that you can see the whole picture rather than only the two numbers you asked for, and the note underneath explains the reasoning step by step.
Blank means unknown. That is the entire input convention, and it removes the usual awkwardness of tools that make you tick boxes or select a target variable from a dropdown. If you fill in four boxes the calculator refuses to guess which one you meant to leave out; if you fill in two it tells you that the system is under-determined and names how many more values it needs. Both messages are more useful than an answer.
One case deserves special mention. The equation v² = u² + 2as is quadratic in velocity, so it has two roots and both are physically meaningful. An object thrown upwards that returns to its starting height has the same speed downwards as it had upwards, and the algebra genuinely cannot tell you which one you meant. The tool defaults to the root that keeps the sign of whichever velocity you already gave it, which is right in the large majority of cases, and lets you override that choice explicitly.
How to Use It
- Choose a positive direction and commit to it. If upwards is positive then gravity is −9.81 m/s². If downwards is positive it is +9.81. Mixing the two mid-problem is the single most common source of wrong SUVAT answers.
- Fill exactly three boxes. Leave the two you are solving for completely empty. A zero is a value, not a blank — typing 0 into the velocity box tells the calculator the object starts at rest, which is very different from not knowing.
- Use consistent units. Everything here is SI: metres, metres per second, metres per second squared, seconds. If your data is in km/h or mph, run it through the speed converter first, and use the acceleration converter for g-forces or ft/s².
- Read the equation name. The subtitle under the headline result names the equation that was applied. That is what you write in an exam script, and it is also how you check the tool has interpreted your inputs the way you intended.
- Set the root branch if a square root appears. When the note tells you a quadratic was solved, decide whether the object is moving forwards or backwards at that instant and set the branch to match.
The Five Equations and Which One Gets Used
All five follow from the definition of constant acceleration, and each omits one variable:
- v = u + at — omits s. Use when displacement is neither known nor wanted.
- s = ut + ½at² — omits v. The workhorse for projectile and free-fall problems.
- s = ½(u + v)t — omits a. Displacement is average velocity multiplied by time, which is exact when acceleration is uniform.
- v² = u² + 2as — omits t. The one to reach for whenever time is not mentioned in the question at all.
- s = vt − ½at² — omits u. Less commonly taught, but it is the natural choice when you know where the object ended up and how fast it was going.
Take the free-fall default loaded above. You know u = 0, a = 9.81 m/s² and t = 3 s, and you want v and s. Time is known, so the fourth equation is not needed; v is unknown, so the second applies for displacement and the first for velocity. v = 0 + 9.81 × 3 = 29.43 m/s, and s = 0 × 3 + ½ × 9.81 × 9 = 44.145 m. The OpenStax University Physics section on motion with constant acceleration derives the same set from first principles if you want to see where they come from rather than take them on trust.
When Three Values Are Not Enough
There is a widely repeated claim that any three SUVAT quantities determine the other two. It is almost true, and the exception is worth knowing. If you supply u, v and a, the tool must find time from t = (v − u) / a. When a is zero, that division is undefined. If u and v are equal, the motion is consistent with any duration at all — an object cruising at constant speed has been doing so for an unknown length of time, and no equation can recover it. If u and v differ while a is zero, the inputs are flatly contradictory: velocity cannot change without acceleration.
The same thing happens with s, u and v when u + v = 0. That describes an object whose final velocity exactly reverses its initial velocity, such as a ball bouncing back at the same speed. Average velocity is zero, so displacement must be zero, and if you have given a non-zero s the inputs are impossible. If you gave s = 0 as well, the situation is real but under-determined; the duration depends on an acceleration you have not supplied.
A third failure mode is a negative discriminant. Solving v² = u² + 2as requires u² + 2as to be zero or positive. A car travelling at 30 m/s decelerating at 6 m/s² cannot cover 100 m, because it stops after 75 m. Ask for the velocity at 100 m and the algebra returns the square root of a negative number. The honest answer is that the scenario does not occur, and this calculator says exactly that rather than quietly returning NaN or zero.
Sign Conventions: Where Most Mistakes Actually Happen
SUVAT is one-dimensional vector algebra, and every one of the five quantities except time carries a sign. Displacement is not distance travelled: a ball thrown up 20 m and caught again has travelled 40 m but has a displacement of zero. Feed a distance into the displacement box and the answers will be wrong in a way that looks superficially reasonable, which is the worst kind of wrong.
The value used for gravity is itself worth stating. Standard gravity is 9.806 65 m/s², and NIST lists it as an exact defined figure rather than a measurement, because it is a conventional reference rather than the gravity at any particular place. Local gravity varies by a few tenths of a per cent with latitude and altitude, which is why 9.81 is a perfectly adequate rounding for schoolwork.
The convention itself is arbitrary but must be consistent. For a projectile it is usually cleanest to make upwards positive, which puts a at −9.81 m/s² and makes the descending velocity negative. For a vehicle braking in a straight line, make the direction of travel positive, which puts deceleration at a negative value. The braking preset above uses u = 30, v = 0, a = −6, and returns s = 75 m and t = 5 s. Enter a = +6 by mistake and you get a displacement of −75 m, which is the calculator telling you the car would have had to reverse.
Time is the one quantity that cannot be negative in a forward-running problem. If a solution produces a negative t, it usually means the described event happened before the moment you called zero — for example asking when a projectile launched upwards was at ground level, where one root is the launch and the other is in the past. This tool reports negative roots rather than hiding them, and flags them in the note.
Where SUVAT Stops Working
Constant acceleration is a strong assumption, and it is the only assumption these equations make. Any situation where the net force changes during the interval falls outside their scope. Air resistance grows with speed, so a skydiver's acceleration falls from 9.81 m/s² towards zero as they approach terminal velocity; SUVAT will overestimate speed badly over a long fall. A spring pulls harder the further it is stretched, which is why oscillating systems need the simple harmonic motion calculator instead. A rocket loses mass as it burns fuel, so even a constant thrust produces rising acceleration.
The practical workaround is segmentation. A train that accelerates for 40 s, cruises for 200 s and then brakes for 30 s is three separate SUVAT problems joined end to end, where the final velocity of one segment becomes the initial velocity of the next. Total displacement is the sum of the three. This is exactly how motion graphs are read, and it is why a velocity-time graph with straight-line segments is equivalent to a sequence of SUVAT calculations — the area under each segment is the displacement.
Two-dimensional motion is a different case. It is not that SUVAT fails, but that you must apply it twice: once horizontally, where acceleration is normally zero, and once vertically, where it is gravity. The two are linked only by sharing the same value of t. If that is your problem, the projectile motion calculator handles the decomposition for you.
How This Differs From the Adjacent Tools
Three Arb Digital tools sit close to this one and it is worth being clear about the boundaries. The velocity calculator answers two specific single-step questions — average velocity from displacement over time, and final velocity from u, a and t — and is the faster choice when that is all you need. This page is the general solver: it handles all ten input combinations and names the equation. The speed converter does no physics at all; it rescales one speed unit into another, which is a units job rather than a kinematics one.
Beyond kinematics, once you have velocity you can carry it into the momentum calculator for p = mv, the work calculator for the energy transferred by the accelerating force, or the potential energy calculator when a height change is involved. For the deceleration case specifically, the stopping distance calculator adds the reaction-time component that pure SUVAT ignores.
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Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using distance where displacement is required — the two differ whenever the motion reverses, and SUVAT is defined in terms of displacement only.
- Flipping the sign convention mid-problem — if up is positive at the start it must stay positive when the object comes back down. Gravity does not change sign at the top of the flight.
- Typing 0 for a value you do not know — blank means unknown, zero means genuinely zero. Entering 0 for a final velocity you were meant to solve for will produce a confident wrong answer.
- Applying SUVAT across a change in force — the equations are invalid over any interval where acceleration is not constant. Split the motion at the point the force changes.
- Ignoring the second root — v² = u² + 2as has two solutions and both can be physically real. Decide which one your scenario describes rather than accepting whichever appears first.
Related Free Tools From Arb Digital
For single-step kinematics use the velocity calculator; for two-dimensional flight use the projectile motion calculator. Follow the motion into dynamics with the momentum calculator and the work calculator, or into oscillation with the simple pendulum calculator. Unit work is handled by the speed converter, the acceleration converter and the force converter, and the full free online tools hub lists everything Arb Digital has published.
Frequently Asked Questions
It is an acronym for the five symbols the equations connect: s for displacement, u for initial velocity, v for final velocity, a for acceleration and t for time. The name is common in British and Commonwealth physics teaching; elsewhere the same relationships are simply called the kinematic equations of motion.
Because a blank is how the calculator identifies what you want solved. Each of the five equations omits exactly one variable, so the tool needs to know which two are unknown before it can choose one. If you fill in four boxes it cannot tell which value you intended to derive.
Yes. If you supply u, v and a with a equal to zero, time cannot be recovered because velocity never changes. If you supply s, u and v where u and v sum to zero, the duration is undetermined. The calculator names the problem instead of returning a number.
Because v squared equals u squared plus 2as is quadratic in velocity, so it has a positive and a negative root and both are physically valid. A ball returning to its launch height has the same speed in the opposite direction. Choose the root that matches the direction of travel at that instant.
No. Constant acceleration is the only assumption they make, and it is a strict one. Air resistance, springs and rockets all violate it. The usual fix is to split the motion into intervals where acceleration is effectively constant and solve each interval separately.
The standard acceleration due to gravity is 9.80665 metres per second squared by definition, usually rounded to 9.81. Its sign depends on the direction you have chosen as positive: negative if upwards is positive, positive if downwards is positive.
No. Displacement is the straight-line change in position including direction, while distance is the total length of the path. An object that goes out and comes back has a large distance and zero displacement. SUVAT is defined entirely in terms of displacement.
This tool is provided for educational and study use. It solves the constant-acceleration equations as written and is not a substitute for engineering analysis or safety assessment of any real moving system.
