How to use it
- Type the three values you know into their boxes, for example Starting speed u 0, Acceleration a 2 and Time t 10. Leave the other two boxes empty.
- Read Number of solutions. Two solutions appear when a quadratic equation has two roots.
- Check the values in Values of the motion: displacement, starting and final speed, acceleration and time.
- Open How it is worked out to see which equation gave each value and what it had to solve.
The five equations
For motion in a straight line with a constant acceleration there are five equations that link the displacement s, the starting speed u, the final speed v, the acceleration a and the time t. Each equation leaves out one of the five values, which is why three known values are enough to find the other two.
(1) v = u + at no s
(2) s = ut + ½at² no v
(3) s = ½(u + v)t no a
(4) v² = u² + 2as no t
(5) s = vt − ½at² no u- Equation (1) is the definition of acceleration: the speed changes by a each second.
- Equation (4) is the one without time, so it is the easiest to use when the time is not given.
- Each pair of equations gives the same motion. The calculator checks every answer against all five, so an answer that fits only one equation is dropped.
How two unknowns are found
The calculator first takes the unknown that an equation holds on its own, with the second unknown left out. If that equation is a quadratic, it has two roots when the discriminant is positive, and one when it is zero. Then the second unknown comes from an equation that is linear in it.
A square root has a plus and a minus sign. When the unknown is squared, as the speed is in equation (4), both signs are kept, and the negative one is checked against the other equations like the positive one.
What the calculator does not cover
- Motion with an acceleration that changes with time, such as a spring or a rocket whose thrust falls. The equations hold only when the acceleration is constant.
- Motion in two or three dimensions: a projectile follows two straight-line motions, which the projectile calculator works out.
- Air resistance and any other force. The equations describe the motion that a force-free model gives.
Worked example: s = 10 m, u = 0 and a = 2 m/s²
The three values given are s, u and a, so v and t are unknown. Equation (4) has no t: v² = u² + 2as = 0 + 2 × 2 × 10 = 40, so v = ±6.3246 m/s. Both signs are kept.
Equation (1) then gives the time: t = (v − u) ÷ a. For v = 6.3246 m/s, t = 6.3246 ÷ 2 = 3.1623 s. For v = −6.3246 m/s, t = −3.1623 s, the moment before the motion started. Equation (2) confirms both: s = ut + ½at² = ½ × 2 × t² = 10 m.
| Solution 1 | Solution 2 | |
|---|---|---|
| Final speed v (m/s) | −6.3246 | 6.3246 |
| Time t (s) | −3.1623 | 3.1623 |
| Displacement s (m) | 10 | 10 |
| Starting speed u (m/s) | 0 | 0 |
| Acceleration a (m/s²) | 2 | 2 |
Frequently asked questions
Which three values do I need?
Any three of the five: displacement s, starting speed u, final speed v, acceleration a and time t. Leave the other two boxes empty. Two values or four are refused, because the equations need exactly three.
Why are there sometimes two answers?
When the unknown is squared, as the speed is in v² = u² + 2as, the square root has a plus and a minus sign. Both are kept, and the calculator checks each one against all five equations before it shows it.
What does a negative time mean?
A negative time is the moment before the motion started, found by running the same constant acceleration backwards. It is a correct answer of the equations, but it is not a moment in an experiment that began at zero.
Can I type km/h and minutes?
Yes. A speed can be written in km/h, so 36 km/h is 10 m/s, and a time in min or h. The calculator converts them before it solves the equations, and the steps use metres per second and seconds.
Is the acceleration the same as g?
Near Earth's surface a falling body has an acceleration of about 9.8 m/s², and the standard value is 9.80665 m/s². The free-fall calculator uses that value. Here a is any constant acceleration you enter, positive or negative.
Does this include air resistance?
No. The equations assume the acceleration stays the same, and air resistance changes it, so the results describe motion without drag. For a fall with no drag, use the free-fall calculator.
Sources
- Wikipedia: Equations of motion (the SUVAT equations)
- Wikipedia: Standard gravity (9.80665 m/s², ISO 80000)
Figures and text last checked on Tuesday, November 10, 2026.