Kinematics Calculator

Five equations, and the skill is picking one

Uniform acceleration links five quantities: initial velocity u, final velocity v, acceleration a, displacement s and time t. Any three of them fix the motion, so the other two are forced. There are five standard equations, and each one leaves out exactly one quantity.

• v = u + at has no s
• s = ut + at²/2 has no v
• v² = u² + 2as has no t
• s = (u + v)t/2 has no a
• s = vt − at²/2 has no u

That pattern is the whole method. Look at what you know and what you want, find the quantity that appears in neither list, and use the equation that omits it. It will have one unknown, which means no simultaneous equations and no rearranging two formulas into each other. The working above names the equation it chose each time, because that choice is where marks are actually lost.

Worked example: dropped from 45 m

Take u = 0, a = 9.81 m/s² and s = 45 m, with downwards positive. Time is wanted and final velocity is not known, so the equation without v applies: 45 = 0 × t + 4.905t², giving t = √(45 / 4.905) = 3.0289 s. Now four quantities are known, so v comes from the simplest equation left: v = 0 + 9.81 × 3.0289 = 29.714 m/s.

Note what did not happen. Nobody had to guess an intermediate value or work in two stages with a rounded number carried between them. Solving for the quantity the question omits first, then finishing with any equation that fits, keeps the full precision through to both answers.

Signs are a choice, and then they are not

Displacement, velocity and acceleration are vectors, so in one dimension each one carries a sign. Which direction is positive is entirely yours to pick, but once picked it applies to all five quantities. Mixing frames halfway through is the most common way a SUVAT answer comes out with the wrong sign or an impossible negative time.

For a dropped object, calling downwards positive keeps everything positive and is the easier arithmetic. For anything thrown upwards, calling upwards positive is more natural, which makes a = −9.81 m/s² throughout, including while the object is still rising. Gravity does not change direction at the top of the flight; only the velocity does.

When there really are two answers

Two of the five equations can produce genuine ambiguity, and this calculator reports both answers when both are possible rather than choosing quietly.

Solving s = ut + at²/2 for time is a quadratic. Throw a ball up at 20 m/s and ask when it is 15 m high: it is there at 0.9907 s on the way up and again at 3.0868 s on the way down. Both are real. Which one the question wants depends on whether it asked for the first time or the last, and that is information the algebra does not contain.

Solving v² = u² + 2as for a velocity takes a square root, so the equation alone leaves the sign open, and the time each root implies settles it. A car that comes to rest in 25 m while decelerating at 2 m/s² gives u² = 100, but only 10 m/s is possible: −10 m/s would need the motion to last −5 s. A ball that lands 5 m below the hand at 10 m/s is different. Taking upwards as positive, it was thrown up at 1.3784 m/s or down at 1.3784 m/s, landing 1.1599 s or 0.87886 s later, and only the question can say which.

What uniform acceleration excludes

Every equation here assumes acceleration is constant over the whole interval. That is a good description of free fall over short distances, of a trolley on a ramp and of steady braking. It is not a description of a car pulling away in traffic, of a falling body that has reached terminal velocity, or of a spring, where the acceleration changes continuously with position.

Air resistance is the usual reason a real measurement disagrees with these numbers. It grows with speed, so the error is small for a dense object over a short drop and large for anything light or any fall over a few seconds. If the numbers matter, the projectile simulator lets you switch drag on and watch how far the ideal answer drifts from the realistic one.

Common mistakes

• Using distance where the equations mean displacement. A ball thrown up 20 m and caught again has travelled 40 m but has a displacement of zero. Every s in these equations is the displacement.
• Making a positive on the way up and negative on the way down. Acceleration due to gravity has one value for the whole flight. Splitting the motion into two halves with different signs double-counts the turn.
• Taking t = 0 as the answer to a quadratic. It is a true root, and it describes the instant before anything happened. For a ball back at its starting height the useful root is the later one.
• Assuming the positive square root. An object moving in the negative direction has a negative velocity even though v² is positive, which is why both roots are checked, and both are shown when both are possible.
• Mixing units. A speed in km/h with a time in seconds is out by a factor of 3.6. Convert everything to metres and seconds before substituting, or use the unit converter first.
• Using g = 10 without saying so. It is a reasonable approximation that shifts answers by about two percent, which is enough to lose a mark when the expected answer used 9.81. The acceleration field is editable for exactly this reason.