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Kinematics (SUVAT) Calculator

Solve constant-acceleration problems with the SUVAT equations. Enter any three of displacement, initial velocity, final velocity, acceleration and time, in the units you like, and the calculator finds the other two. When a problem has two valid answers, such as a ball passing the same height on the way up and on the way down, both are shown.

Physics · MotionUpdated Sources cited

Enter any three quantities

Leave the unknowns blank. Use negative values for quantities pointing opposite to your chosen positive direction.

Examples:

Result

QuantityValue
s displacement(given)50 m
u initial velocity(given)20 m/s
v final velocity(given)0 m/s
a acceleration−4 m/s²
t time5 s
Assumptions behind this result
  • The acceleration is constant throughout the motion.
  • Motion is along a straight line. Signs follow the positive direction you chose; negative values point the opposite way.

Step-by-step solution

  1. List the known quantities

    Write down what is known and what is to be found, in SI units.

    s=50 m,u=20 m/s,v=0 m/s,a=?,t=?s = 50\ \text{m},\quad u = 20\ \text{m/s},\quad v = 0\ \text{m/s},\quad a = ?,\quad t = ?

  2. Find the time

    Use the equation that does not involve the acceleration, which is not needed here.

    s=12(u+v)t⇒t=2(50)20+0=5 ss = \tfrac{1}{2}(u + v)t \Rightarrow t = \frac{2(50)}{20 + 0} = 5\ \text{s}

  3. Find the acceleration

    Use the equation that does not involve the time, which is not needed here.

    v2=u2+2as⇒a=(0)2−(20)22(50)=−4 m/s2v^2 = u^2 + 2as \Rightarrow a = \frac{(0)^2 - (20)^2}{2(50)} = -4\ \text{m/s}^2

What the calculator does

Give it any three of the five SUVAT quantities and it:

  • picks the one equation that contains your three values and the unknown you need;
  • rearranges and solves it, showing each substitution;
  • reports both solutions when a quadratic has two valid roots;
  • converts units for you, so you can mix km/h, m/s and ft/s.

The five equations

Each equation leaves out exactly one quantity, shown in brackets:

v=u+at(s)v = u + at \quad (s)
s=ut+12at2(v)s = ut + \tfrac{1}{2}at^2 \quad (v)
s=vt−12at2(u)s = vt - \tfrac{1}{2}at^2 \quad (u)
s=12(u+v)t(a)s = \tfrac{1}{2}(u + v)t \quad (a)
v2=u2+2as(t)v^2 = u^2 + 2as \quad (t)

To find two unknowns, the calculator uses the equation without the second unknown first, then any equation for the second.

Worked example 1: a stone dropped from 20 m

Take down as positive. The stone starts at rest, so u = 0, a = 9.80665 m/s² and s = 20 m.

  1. Final speed, from v² = u² + 2as: v = √(2 × 9.80665 × 20) = 19.81 m/s.
  2. Time, from s = ut + ½at²: t = √(2 × 20 / 9.80665) = 2.02 s.

Worked example 2: two answers

A ball is thrown straight up at 15 m/s. When is it 10 m above the launch point? Take up as positive: u = 15 m/s, a = −9.80665 m/s², s = 10 m.

From s = ut + ½at², 10 = 15t − 4.903t². This quadratic has two roots, t = 0.98 s and t = 2.08 s. The ball is at 10 m on the way up, moving at +5.37 m/s, and again on the way down, moving at −5.37 m/s. Both answers are correct.

Worked example 3: braking

A car slows from 25 m/s to rest in 40 m. With u = 25, v = 0 and s = 40, v² = u² + 2as gives a = −7.81 m/s². The negative sign means the acceleration is opposite to the motion. The stop takes t = 2s/(u + v) = 3.2 s.

Common mistakes

  • Using g = +9.81 m/s² for an object moving upwards when up is positive. Gravity points down, so a = −9.81 m/s².
  • Mixing units, such as km/h with seconds. The calculator converts for you; on paper, change everything to SI first.
  • Throwing away a root without checking it. A negative time is usually not physical, but two positive roots often both are.
  • Using SUVAT for changing acceleration, such as a falling object with air resistance.

Practical applications

  • Road safety. Stopping distances and braking forces.
  • Sport. Sprint starts, jumps and throws.
  • Engineering. Lifts, conveyors and vehicle performance.
  • Exams. SUVAT problems are a core part of GCSE, A-level, AP and introductory university physics.

Assumptions and limitations

The motion is in a straight line with constant acceleration. For motion in two dimensions, treat the horizontal and vertical directions separately, as the projectile motion calculator does. The default value for g is the standard 9.80665 m/s²; local gravity varies by about ±0.3% across the Earth’s surface. Results are computed in double precision and shown to 6 significant figures.

Frequently asked questions

What does SUVAT stand for?

The five quantities in constant-acceleration motion. s is displacement, u initial velocity, v final velocity, a acceleration and t time. Each of the five SUVAT equations leaves out one of them.

Why do I sometimes get two answers?

Equations with t² or v² can have two solutions. A ball thrown upwards passes the same height twice, once rising and once falling. Both are physically valid, so the calculator shows both and you choose the one that fits your situation.

Which direction is positive?

You choose. Pick a positive direction and keep to it. For something thrown upwards with up as positive, u is positive and a = −9.81 m/s². A negative answer for v or s means the motion or position is in the negative direction.

Can I enter more than three values?

Yes. With four or five values the calculator checks that they are consistent with each other and tells you if they are not, which is a good way to check homework answers.

Does this work if the acceleration changes?

No. The SUVAT equations are only valid when the acceleration is constant. For changing acceleration you need calculus, integrating a(t) to get v(t) and s(t).

References

  1. Young, H. D. & Freedman, R. A. (2019). University Physics (15th ed.), chapter 2. Pearson.
  2. Halliday, D., Resnick, R. & Walker, J. (2018). Fundamentals of Physics (11th ed.), chapter 2. Wiley.
  3. BIPM (2019). The International System of Units (SI), 9th edition: standard acceleration of gravity.

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