The five SUVAT equations
For motion in a straight line with constant acceleration, five quantities are linked: displacement s, initial velocity u, final velocity v, acceleration a and time t. Know any three and the other two follow.
Each equation leaves out one quantity, so the calculator picks the one that does not need the value you are missing.
| Equation | Leaves out |
|---|---|
| v = u + at | s |
| s = ut + ½at² | v |
| s = ½(u + v)t | a |
| v² = u² + 2as | t |
| s = vt − ½at² | u |
Worked example
A car speeds up from 10 m/s to 30 m/s at 4 m/s². Time t = (v − u) ÷ a = (30 − 10) ÷ 4 = 5 s. Distance s = (v² − u²) ÷ 2a = (900 − 100) ÷ 8 = 100 m.
Check with s = ½(u + v)t = ½ × 40 × 5 = 100 m. The average velocity over the interval is (10 + 30) ÷ 2 = 20 m/s.
When there are two answers for time
Solving for time from s, u and a (or s, v and a) means solving a quadratic, which can have two non-negative roots. A ball thrown straight up at 20 m/s (a = −9.80665 m/s²) is 15 m high at about 0.99 s on the way up and again at about 3.09 s on the way down.
The calculator shows the earlier time as the main result and lists both solutions in a table. If no real, non-negative time exists, the object never reaches that displacement and you get a clear message instead.
Speed, distance and time in everyday units
Average velocity mode accepts distance in meters, kilometers, miles or feet, time in seconds, minutes, hours or h:mm:ss, and speed in m/s, km/h or mph. The calculator converts every input to SI, applies v̄ = s ÷ t, s = v̄t or t = s ÷ v̄, and converts the answer back, showing each conversion as a step.
Example: a marathon of 42.195 km run in 2:00:00 is 42,195 m ÷ 7,200 s = 5.860 m/s, or 21.0975 km/h. At 60 mph for 1.5 h a car covers 60 × 1,609.344 ÷ 3,600 = 26.8224 m/s × 5,400 s = 144,841 m, which is 90 miles. Unit factors are exact: 1 mi = 1,609.344 m and 1 ft = 0.3048 m.
Signs, units and average velocity
Choose one direction as positive and give every velocity, acceleration and displacement a sign relative to it. Deceleration is simply a negative acceleration when moving in the positive direction.
Average velocity mode uses v̄ = Δs ÷ Δt, which holds for any motion, and can also rearrange to s = v̄t or t = s ÷ v̄. The shortcut v̄ = (u + v) ÷ 2 is only true when acceleration is constant. SUVAT mode works in SI units: meters, seconds, m/s and m/s².