Displacement Calculator (s = ut + ½at²)
Solve the SUVAT equation s = ut + ½at² for displacement, initial velocity, acceleration, or time — with worked steps shown.
The second SUVAT equation
For motion with constant acceleration, the displacement after time t combines steady motion at the initial velocity with the extra distance added by acceleration:
- s — displacement, in metres (m)
- u — initial velocity, in metres per second (m/s)
- a — acceleration, in metres per second squared (m/s²)
- t — time elapsed, in seconds (s) — always positive
Enter any three values and leave the fourth blank; the calculator solves for it and shows the substitution step.
Worked example
A car starts from rest (u = 0 m/s) and accelerates at 2 m/s² for 5 seconds. How far does it travel?
- Formula: s = ut + ½at²
- Substitute: s = 0 × 5 + ½ × 2 × 5²
- Result: s = 25 m
Solving in reverse
The same formula rearranges to find initial velocity or acceleration directly, and time by solving a quadratic:
- Initial velocity: u = (s − ½at²) / t — enter s, a, and t, leave u blank.
- Acceleration: a = 2(s − ut) / t² — enter s, u, and t, leave a blank.
- Time: solve ½at² + ut − s = 0 — enter s, u, and a, leave t blank. The calculator picks the positive root.
For example, a car reaching 25 m from rest with acceleration 2 m/s² takes t = 5 s.
Common mix-ups
- Using the wrong sign for s. Displacement measured against your chosen positive direction can come out negative — that's a valid answer, not an error.
- Skipping the ½ on the acceleration term. s = ut + at² (without the ½) is a common slip that doubles the acceleration contribution.
- Expecting a speed, not a distance. s = ut + ½at² gives how far the object moved — see the final velocity calculator (v = u + at) for how fast it ends up.
Where this shows up
s = ut + ½at² is the workhorse equation for stopping distances, free-fall drop height, and projectile range along one axis. Once displacement is known, it connects to work (W = Fs) done by a constant force over that distance, and to potential energy (PE = mgh) when the displacement is a height.
Frequently asked questions
- What is the formula for displacement with constant acceleration?
- The second SUVAT equation: s = ut + ½at², where s is displacement (m), u is initial velocity (m/s), a is acceleration (m/s²), and t is elapsed time (s).
- How do I find time from displacement, initial velocity, and acceleration?
- Solving for time means rearranging s = ut + ½at² into the quadratic ½at² + ut − s = 0 and taking the physically meaningful root. The calculator does this automatically — enter s, u, and a, leave t blank.
- Why does solving for time sometimes give "no real solution"?
- The quadratic for t has a discriminant u² + 2as. If that value is negative, there is no real time at which the object reaches that displacement with those starting conditions — the calculator reports this instead of an invalid number.
- Can displacement or initial velocity be negative?
- Yes. A negative displacement means net movement opposite to the direction you called positive, and a negative initial velocity means starting out moving that way. Only time must stay positive.
- What units does this calculator use?
- SI units throughout — metres (m) for displacement, metres per second (m/s) for velocity, metres per second squared (m/s²) for acceleration, and seconds (s) for time.
- How is this different from the final velocity calculator?
- s = ut + ½at² gives the distance travelled over a known time; it doesn't directly give the speed at the end. Use the final velocity calculator (v = u + at) when you need the velocity instead of the distance.