SUVAT Calculator takes any three of s, u, v, a and t and solves the other two, showing which equation it picked, the rearrangement, the substituted numbers and a check on all five.
Fill in any three values and leave two blank. Use a minus sign for the opposite direction.
Step-by-Step Working
- Known values in SI units: s = 50 m, u = 0 m/s, a = 9.81 m/s²
- To find: final velocity (v) and time taken (t)
- Pick the equation without t: v² = u² + 2as, rearranged to v = ±√(u² + 2as)
- Substitute: v = ±√(0² + 2 × 9.81 × 50) = 31.32 m/s
- Then use v = u + at, rearranged to t = (v − u) ÷ a
- Substitute: t = (31.32 − 0) ÷ 9.81 = 3.19 s
- Result: v = 31.32 m/s and t = 3.19 s
All Five Equations Checked
| Equation | Left Side = Right Side | Role |
|---|---|---|
| v = u + at | 31.32 = 31.32 | Used to solve |
| s = ½(u + v)t | 50 = 50 | Agrees |
| s = ut + ½at² | 50 = 50 | Agrees |
| s = vt − ½at² | 50 = 50 | Agrees |
| v² = u² + 2as | 981 = 981 | Used to solve |
Motion Graphs
How the SUVAT Calculator Finds Two Unknowns
The SUVAT Calculator works on motion in a straight line with constant acceleration. That motion has five quantities, written s, u, v, a and t. Enter any three, leave the other two boxes blank, and it solves both missing values.
It first picks the one equation that leaves out the second unknown, so only a single unknown remains. It then uses a second equation for the other one. The step-by-step panel shows both equations, each rearrangement and the numbers substituted in.
After solving, the SUVAT Calculator puts the complete set of values back into all five equations and shows that the two sides match. The result cards add the average velocity, the velocity at half the distance and the same answers in a second unit.
The Five SUVAT Equations
Each of the five SUVAT equations links four of the quantities and leaves one out. The quantity left out is the key to choosing. Find the one your question neither gives nor asks for, then use the equation without it.
| Equation | Leaves out | Typical use |
|---|---|---|
| v = u + at | s | Velocity after a known time |
| s = ½(u + v)t | a | Displacement when both velocities are known |
| s = ut + ½at² | v | Displacement from a start velocity and a time |
| s = vt − ½at² | u | Displacement when only the final velocity is known |
| v² = u² + 2as | t | Velocity or displacement with no time given |
Here s is displacement in meters, u is initial velocity and v is final velocity in m/s, a is acceleration in m/s² and t is time in seconds. The SUVAT Calculator also takes feet, miles, km/h, mph, ft/s², g and minutes, and converts them to SI units before solving.
For SUVAT without v, use the third row, s = ut + ½at². In the SUVAT Calculator you simply leave the final velocity box empty. The same applies to any other letter your question does not mention.
The name is British. US textbooks such as OpenStax University Physics call them the kinematic equations and write v0 for u and x − x0 for s. The physics is identical, so the calculator suits either notation.
SUVAT Rearranged for t and Other Unknowns
The equations are usually printed with v or s as the subject. The table gives them rearranged for the unknown you need. These are the same forms the SUVAT Calculator prints in its working before it substitutes your numbers.
| To find | Known values | Rearranged form |
|---|---|---|
| t | u, v, a | t = (v − u) ÷ a |
| t | s, u, v | t = 2s ÷ (u + v) |
| a | u, v, t | a = (v − u) ÷ t |
| a | s, u, v | a = (v² − u²) ÷ (2s) |
| a | s, u, t | a = 2(s − ut) ÷ t² |
| s | u, v, a | s = (v² − u²) ÷ (2a) |
| u | v, a, t | u = v − at |
| u | s, a, t | u = (s − ½at²) ÷ t |
| v | s, u, t | v = 2s ÷ t − u |
| v | s, u, a | v = ±√(u² + 2as) |
Time from s, u and a is the awkward case, because s = ut + ½at² is a quadratic in t. Solving it with the quadratic formula gives the expression below, which can produce two different times.
$$t = \frac{-u \pm \sqrt{u^2 + 2as}}{a}$$
The SUVAT Calculator reaches the same answers by a shorter route. It finds v from v² = u² + 2as first, then t from t = (v − u) ÷ a. For a stone dropped 50 m from rest, both routes give 3.19 seconds.
Minus Signs and Two Valid Answers
Displacement, velocity and acceleration all have a direction. Choose one direction as positive before you type anything, and give every value that points the other way a minus sign. A ball thrown upward at 20 m/s has u = 20 and a = −9.81.
That ball passes a height of 10 m twice, and the SUVAT Calculator reports both. It is at 10 m after 0.5835 s, rising at 14.28 m/s, and again after 3.49 s, falling at −14.28 m/s. The main results use the first time and the last card shows the second.
Ask for a height of 25 m and there is no answer, because the square root goes negative. The calculator says the ball never reaches that displacement and turns back at 20.39 m. That figure is the peak height, which the Maximum Height Calculator finds for any launch angle.
When the velocity changes sign during the motion, a card shows the turnaround time, the farthest point and the total distance travelled. Distance and displacement are different numbers in that case. The Displacement Calculator works on displacement alone, including trips made of several legs.
SUVAT Questions Worked With Steps
The example menu loads four standard SUVAT questions with their numbers already filled in. Pick one to see the method, then overwrite the values with your own. The answers below are the ones the SUVAT Calculator returns.
| Question | Values entered | Values solved |
|---|---|---|
| Stone dropped 50 m | s = 50 m, u = 0, a = 9.81 m/s² | v = 31.32 m/s, t = 3.19 s |
| Car braking to a stop | u = 30 m/s, v = 0, a = −6 m/s² | s = 75 m, t = 5 s |
| Ball thrown straight up | s = 10 m, u = 20 m/s, a = −9.81 m/s² | v = 14.28 m/s at 0.5835 s, or −14.28 m/s at 3.49 s |
| 0 to 60 mph in 4 seconds | u = 0, v = 60 mph, t = 4 s | a = 0.6838 g, s = 176 ft |
Take the stone. No time is given, so the working starts with v² = u² + 2as. That gives v = √(2 × 9.81 × 50) = 31.32 m/s, or 112.76 km/h. Then t = (31.32 − 0) ÷ 9.81 = 3.19 s. Downward is positive here, so every value stays positive.
The cards add detail that a textbook answer leaves out. The stone is already moving at 22.15 m/s at the 25 m mark, which it passes after 2.26 s. After half the time it has fallen only 12.5 m, a quarter of the drop. The Free Fall Calculator handles drops like this from a single known value.
The braking car shows the same pattern in reverse. It stops in 75 m, yet at 37.5 m it is still traveling at 21.21 m/s, which is 71 percent of its starting speed. Most of the speed is lost in the second half of the stopping distance.
For a SUVAT calculator PDF of your working, press Print Results and choose Save as PDF in the print dialog. The page prints with the step-by-step working, the five-equation check and the motion graphs all open.
Deriving SUVAT From a Velocity-Time Graph
With constant acceleration, velocity plotted against time is a straight line from u to v. The gradient of that line is the acceleration, so a = (v − u) ÷ t. Rearranged, that is the first equation, v = u + at.
The area under the line is the displacement. The shape is a trapezoid with parallel sides u and v and width t, so s = ½(u + v)t. The Motion Graphs panel in the SUVAT Calculator shades this area for your own numbers.
The other three come from substitution. Put v = u + at into the area formula to get s = ut + ½at². Put u = v − at into it to get s = vt − ½at². Replace t with (v − u) ÷ a to get v² = u² + 2as.
Quick SUVAT Answers
Why is it called SUVAT?
The name is the five symbols read in a row. They stand for displacement s, initial velocity u, final velocity v, acceleration a and time t. The label is used mainly in British schools, and the same formulas are called the kinematic equations or the equations of motion elsewhere.
Is SUVAT in A-level maths?
Yes, it sits in the mechanics part of A-level Mathematics. The AQA specification asks students to understand, use and derive the formulae for constant acceleration for motion in a straight line. The full A-level then extends them to two dimensions using vectors.
Two-dimensional SUVAT is projectile motion, with one set of equations for the horizontal direction and one for the vertical. The Projectile Motion Calculator handles that case, including the range, flight time and peak height of a launch.
Is SUVAT only for constant acceleration?
Yes. Every equation assumes the acceleration keeps one value for the whole motion, along a straight line. If it changes in clear stages, such as a car speeding up and then braking, treat each stage as its own SUVAT problem with its own acceleration.
Air resistance breaks the assumption on long falls, because drag grows with speed until the object stops accelerating at its terminal velocity. The Free Fall with Air Resistance Calculator models that kind of fall.
Is acceleration always 9.8 in physics?
No. A value near 9.8 m/s² applies only to objects falling freely close to the Earth’s surface. Standard gravity is defined as 9.80665 m/s², and school questions round it to 9.81 or 9.8. The g unit in the SUVAT Calculator uses the standard value.
Every other motion has its own acceleration. The braking car above has −6 m/s², and a car that reaches 60 mph in 4 seconds has 6.71 m/s². When acceleration is the unknown, leave its box blank and enter three of the other values.
How far will a free-falling object fall in 10 seconds?
From rest and with no air resistance, s = ½ × 9.81 × 10² = 490.5 m, which is about 1,609 ft. Its velocity by then is 98.1 m/s, or 219 mph. Enter u = 0, a = 9.81 and t = 10 in the SUVAT Calculator to see the full working.
The distance grows with the square of the time. Two seconds of free fall covers 19.62 m, and twelve seconds covers 706.32 m. Real falls of that length come out shorter, since drag slows the object on the way down.
What is your acceleration if you go from 0 to 60 mph in 4 seconds?
It is 6.71 m/s², which equals 22 ft/s² or 0.68 g. Convert 60 mph to 26.82 m/s, then use a = (v − u) ÷ t = 26.82 ÷ 4. The distance covered in those 4 seconds is 53.64 m, or 176 ft, from s = ½(u + v)t.
This is an average figure, because a real car does not hold one acceleration all the way to 60 mph. For a full drag-strip run, the Quarter Mile Calculator estimates the elapsed time and trap speed from weight and power.
How do I convert acceleration to velocity?
Multiply the acceleration by the time it acts and add the starting velocity, which is v = u + at. An object that starts at rest and gains 2 m/s every second for 5 seconds ends at 10 m/s. Acceleration alone cannot give a velocity without a time or a distance.