Constant-acceleration problems all share one structure. Five quantities describe the motion, five equations connect them, and any three known values pin down the other two. This calculator does the bookkeeping. You say which three you have, and it picks the equations, solves them in order and shows every substitution, whether for homework, exam review or a quick engineering estimate.
How to use the SUVAT calculator
- In Which three values do you know? choose one of the ten combinations, such as “Know u, a, t → find s and v”.
- Fill in the three fields that appear, chosen from Displacement (s), Initial velocity (u), Final velocity (v), Acceleration (a) and Time (t). Each has its own unit menu.
- In Show results in, pick a unit system: SI (m, m/s, m/s², s), Road (m, km/h, m/s², s), US (ft, ft/s, ft/s², s) or US road (ft, mph, ft/s², s).
- The result lists all five quantities, marked given or calculated, and the work section shows each equation used.
The five SUVAT equations
Each equation leaves out one quantity, so pick the one that omits the quantity you neither know nor need.
| Equation | Leaves out | Single-equation tool |
|---|---|---|
| v = u + at | s | velocity calculator |
| s = ut + ½at² | v | s = ut + ½at² calculator |
| s = vt − ½at² | u | (built into this calculator) |
| s = ½(u + v)t | a | s = ½(u + v)t calculator |
| v² = u² + 2as | t | final velocity calculator |
Five quantities taken three at a time give ten combinations of knowns, and the calculator offers all ten. With u, a and t it gets v from the first equation and s from the second. With s, u and v it uses the fourth and fifth.
Sign-convention checklist
- Choose one positive direction and keep it throughout.
- s, u, v and a are signed. A value pointing the other way gets a minus sign.
- t is always positive. The calculator rejects zero or negative times.
- For free fall with up as positive, use a = −9.80665 m/s² (standard gravity, g₀).
- An object slows down when a and its velocity have opposite signs, whatever the sign of a itself.
- s is net displacement. If u and v have opposite signs the object reversed, and the calculator warns that the distance traveled exceeds |s|.
Worked example
Known: u = 0, a = 2 m/s², t = 10 s (case "u, a, t").
v = u + at = 0 + 2 × 10 = 20 m/s
s = ut + ½at² = 0 × 10 + ½ × 2 × 10² = 100 m
Switch Show results in to Road and v reads 72 km/h. Only the display units change.
A harder case: s, u and a known
Choose “Know s, u, a” with s = 50 m, u = 20 m/s and a = −3 m/s². Picture a puck sent up a long, nearly frictionless icy ramp, so the same acceleration acts going up and coming down. The unknowns, v and t, need two steps.
v² = u² + 2as = 20² + 2 × (−3) × 50 = 100, so |v| = 10 m/s
Solving 50 = 20t − 1.5t² gives t = 3.33 s or t = 10 s
At the first time, v = 20 − 3 × 3.33 = +10 m/s. At the second, v = −10 m/s.
The puck passes the 50 m mark at 3.33 s while still climbing. It stops at about 66.7 m, slides back, and passes 50 m again at 10 s. The calculator reports the first pass and gives the second in a note. Raise s to 80 m and v² becomes −80, which has no real square root. The puck never gets that far, and the calculator says so instead of inventing an answer.
In US road units, “Know u, v, a” with u = 60 mph, v = 0 and a = −20 ft/s² gives a stop in 4.4 s over 193.6 ft.
When SUVAT does not apply
The equations assume acceleration that stays constant in size and direction throughout. They fail when:
- Acceleration changes, as with a car whose engine pulls harder at some speeds, or a rocket losing mass as it burns fuel. Split the motion into stretches of roughly constant acceleration, or use calculus.
- Air drag matters. Drag grows with speed, so a falling skydiver’s acceleration shrinks toward zero at terminal velocity. SUVAT with g alone overestimates the speed.
- The motion leaves a single line. In circular motion the acceleration keeps changing direction. A projectile’s acceleration is constant but its path is two-dimensional, so apply SUVAT to the horizontal and vertical parts separately, as the projectile motion calculator does.
For dropped objects, the free fall calculator applies the same equations with g already filled in.
Frequently asked questions
What does SUVAT stand for?
It lists the five quantities of constant-acceleration motion: s for displacement, u for initial velocity, v for final velocity, a for acceleration and t for time. The name is common in UK and Commonwealth classrooms. US textbooks often write the same equations with Δx, v₀ and v.
Why do I need exactly three values?
Each SUVAT equation links four of the five quantities, and only two of the equations are independent. Two independent equations can fix two unknowns, so three values must be given. With fewer the motion is not determined. With more you risk giving numbers that contradict each other.
How high does a ball thrown straight up at 12 m/s go?
Take up as positive and use the u, v, a case with u = 12 m/s, v = 0 at the top and a = −9.80665 m/s². The calculator gives s = 7.34 m and t = 1.22 s to reach the peak, ignoring air resistance.
Why does the calculator mention a second time?
When displacement is one of the knowns and time is an unknown, the equation for t is quadratic. Both roots can be positive if the object passes the same point twice, going out and coming back. The first time is used for the results and the second is given in a note.