Kinematics Solver — SUVAT Equations
Solve all four SUVAT kinematic equations simultaneously. Enter any 3 of the 5 variables (displacement, initial velocity, final velocity, acceleration, time) and the solver finds the remaining values instantly. No signup, runs entirely in your browser.
⏱ 7 min read · Complete guide below
Enter any 3 known values. Leave unknowns blank.
How to Use the Kinematics Solver
- 1Identify which 3 of the 5 SUVAT variables you know: displacement (s), initial velocity (u), final velocity (v), acceleration (a), time (t).
- 2Enter them in SI units — meters, m/s, m/s², seconds. Leave the two unknowns blank.
- 3Use negative values for anything opposing your chosen positive direction — deceleration is a negative a.
- 4The solver applies whichever SUVAT equations fit and fills in the remaining two variables.
Worked Example: Emergency Stop from 100 km/h
A car travels at 100 km/h — first convert to SI: 100 / 3.6 ≈ 27.8 m/s. The driver brakes hard on dry asphalt, decelerating at 7 m/s² (about 0.7 g, a realistic maximum for road tires). The knowns are u = 27.8, v = 0, a = −7. The solver applies v² = u² + 2as to get the stopping distance: s = u² / (2 × 7) = 772.8 / 14 ≈ 55.2 m, and v = u + at for the stopping time: t = 27.8 / 7 ≈ 4.0 s.
The quadratic term is what surprises people: because s grows with u², a car at 50 km/h (13.9 m/s) needs only 13.8 m to stop — one quarter of the distance at double the speed, not half. And this is pure braking distance; add roughly one second of driver reaction time (another 27.8 m at highway speed) for the true stopping gap. Change u in the solver and watch how steeply the distance climbs.
What SUVAT Is — and Its One Key Assumption
SUVAT is a mnemonic for the five quantities that describe motion in a straight line: s(displacement), u (initial velocity), v (final velocity), a(acceleration), and t (time). The four SUVAT equations link these quantities, and there is one crucial condition for using them: the acceleration must be constant. That is why they work perfectly for free fall (where gravity provides a steady 9.81 m/s²) and for a car braking evenly, but not for situations where the acceleration itself changes over time, such as air resistance building up at speed. Whenever you reach for these equations, the first question to ask is whether the acceleration is genuinely uniform for the phase of motion you are analysing.
Choosing Three Knowns and Picking the Right Equation
The reason the solver only needs three of the five variables is that each SUVAT equation involves four of them, leaving one out. Give it any three and at least one equation contains exactly one unknown, which can be solved immediately; that result then unlocks the rest. The four equations differ by which variable they omit — v = u + at has no displacement, s = ut + ½at² has no final velocity, v² = u² + 2as has no time, and s = ½(u+v)t has no acceleration. When solving by hand, the trick is to pick the equation that contains your three knowns and the one unknown you want, so no intermediate steps are needed. The solver does this selection automatically.
Getting the Signs Right
More kinematics mistakes come from signs than from any other source. Before you start, choose a positive direction — usually the direction of initial motion — and stick to it for the whole problem. Anything pointing the other way is negative: a car decelerating has a negative acceleration, and an object thrown upward has a positive initial velocity but a negative gravitational acceleration. Getting these signs consistent is what lets a single set of equations handle a ball thrown up, a car braking, and an object in free fall without changing the formulas. If a result comes out with an unexpected sign, it usually means the quantity simply points opposite to your chosen positive direction — which is often useful information rather than an error.
Kinematics Tips
Free fall
For free fall, set a = 9.81 m/s² (downward), u = 0, and enter either the height (s) or time (t). The solver finds the impact velocity and time of fall.
Braking distance
For a car braking to rest, set v = 0 and enter initial speed and deceleration (negative a). The solver gives stopping distance (s) and time to stop.
Sign convention
Choose a positive direction (usually the direction of initial motion). Displacement, velocity, and acceleration opposing that direction should be entered as negative values.
Multiple phases
For motion in multiple phases (e.g. accelerating then coasting), solve each phase separately. The final velocity of one phase becomes the initial velocity of the next.
Frequently Asked Questions
What are the SUVAT equations?
SUVAT equations describe motion under constant acceleration. The four equations are: (1) v = u + at, (2) s = ut + ½at², (3) v² = u² + 2as, (4) s = ½(u+v)t. These relate displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).
How many known values do I need?
You need exactly 3 of the 5 variables to solve for the other 2. The calculator automatically identifies which equations can be applied and solves sequentially until all variables are found.
What does SUVAT stand for?
SUVAT is a mnemonic: S = displacement, U = initial velocity, V = final velocity, A = acceleration, T = time. These are the five variables governed by the equations of uniform (constant) acceleration.
Can this handle deceleration?
Yes. Enter acceleration as a negative value for deceleration. For example, a car braking from 30 m/s with a = −5 m/s² will come to rest (v = 0) in 6 seconds over 90 meters.
What if I get a contradictory result?
If the values are physically inconsistent (e.g. final velocity greater than initial with negative acceleration), the solver may not find a valid solution. Check that your values are physically realistic and consistent with constant acceleration.
Is my data stored?
No. All calculations run locally in your browser. No data is sent to any server.
When can I use the SUVAT equations?
Only when acceleration is constant over the phase of motion you are analysing. That covers many common problems — free fall under gravity, a car braking steadily, an object sliding down a fixed incline — but not situations where the acceleration itself changes, such as motion with significant air resistance. If the acceleration is not uniform, the SUVAT equations do not apply and you need calculus-based methods instead.
Which SUVAT equation should I choose for a problem?
Pick the equation that contains your three known values and the single unknown you want, so you can solve in one step. Each equation leaves out one variable: v = u + at omits displacement, s = ut + ½at² omits final velocity, v² = u² + 2as omits time, and s = ½(u+v)t omits acceleration. Identify the variable you do not have and do not need, then use the equation that excludes it. The solver does this automatically.
How do I handle an object thrown straight up?
Choose upward as positive. The initial velocity is positive, and gravity gives a constant acceleration of about −9.81 m/s² (negative because it points down). At the highest point the velocity is momentarily zero, which is a useful known value. Using consistent signs, the same SUVAT equations then give the time to the top, the maximum height, and the time to fall back — no special "up" formulas needed.
Why does my answer have a negative sign?
A negative result usually just means the quantity points opposite to the positive direction you chose at the start. For example, if you take rightward as positive and solve for the velocity of an object moving left, you will correctly get a negative value. It is rarely an error — it is the equations telling you the direction. Only worry if a magnitude that must be positive, like a time of flight, comes out negative, which points to inconsistent inputs.
Can the solver handle motion in two phases?
Solve each phase separately, because the acceleration is usually different in each. Work out the end of the first phase — for instance the velocity at the moment a car stops accelerating — then use that final velocity as the initial velocity for the second phase. Chaining phases this way lets you analyse realistic journeys like accelerate-then-coast or throw-then-fall while keeping each phase within the constant-acceleration assumption.