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SUVAT Equations Calculator and Formulas

The SUVAT equations link displacement s, initial velocity u, final velocity v, acceleration a and time t for motion with constant acceleration: v = u + at, s = ut + ½at², s = vt − ½at², s = ½(u + v)t and v² − u² = 2as. Each one leaves out a single variable, so you choose the equation without the quantity you are neither given nor asked for.

SUVAT equations

v = u + a·t
s = u·t + ½·a·t²
v² = u² + 2·a·s
s = ½·(u + v)·t
s = v·t − ½·a·t²
s
displacement, m
u
initial velocity, m/s
v
final velocity, m/s
a
acceleration, m/s²
t
time, s

Convert every value to SI units (m, s, kg, A, V, Ω) before substituting into the formula. The calculator does this for you and shows the conversion as a step.

25 m
m/s²
sdisplacement
25 m
displacement: 25 m
The velocity–time graph runs from 0 m/s to ? over 5 s. The gradient of the line is the acceleration, 2 m/s², and the shaded area under it is the displacement, 25 m.

Step-by-step solution

  1. Equations = u·t + ½·a·t²
  2. Formulas = u·t + ½·a·t²
  3. Substitutes = 0 m/s · 5 s + ½ · 2 m/s² · (5 s)²
  4. Answers = 25 m

SUVAT examples

  1. Example 1

    A car sets off with an initial velocity of 0 m/s and accelerates uniformly at 2 m/s² for 5 s. Calculate its displacement.

    Given:u = 0 m/sa = 2 m/s²t = 5 s
    1. Equations = u·t + ½·a·t²
    2. Formulas = u·t + ½·a·t²
    3. Substitutes = 0 m/s · 5 s + ½ · 2 m/s² · (5 s)²
    4. Answers = 25 m

  2. Example 2

    A trolley passes the top of a ramp at 3 m/s and accelerates uniformly at 2 m/s². Calculate its velocity after it has travelled 4 m down the ramp.

    Given:u = 3 m/sa = 2 m/s²s = 4 m
    1. Equationv² = u² + 2·a·s
    2. Formulav = u² + 2·a·s
    3. Substitutev = (3 m/s)² + 2 · 2 m/s² · 4 m
    4. Answerv = 5 m/s

  3. Example 3

    A motorcyclist on a straight road speeds up uniformly from 10 m/s to 30 m/s over a distance of 100 m. Calculate how long this takes.

    Given:s = 100 mu = 10 m/sv = 30 m/s
    1. Equations = ½·(u + v)·t
    2. Formulat = 2·su + v
    3. Substitutet = 2 · 100 m10 m/s + 30 m/s
    4. Answert = 5 s

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How to use the SUVAT equations

The SUVAT equations describe motion in a straight line with constant (uniform) acceleration. The name comes from the five quantities: s displacement, u initial velocity, v final velocity, a acceleration and t time. Each equation contains four of the five, so if you know three quantities you can always find the other two. Pick the equation that leaves out the quantity you are neither given nor asked for.

At GCSE you meet v² − u² = 2as, which is printed on the equation sheet, alongside acceleration = change in velocity ÷ time taken. At A-level you use all five, in physics and in the mechanics part of maths. Displacement, velocity and acceleration are vectors, so choose a positive direction first: anything in the opposite direction, such as a braking acceleration or the acceleration due to gravity on a ball thrown upwards, is negative.

An everyday picture: a car joining a motorway or a stone dropped down a well accelerates at a roughly constant rate, so SUVAT gives good answers. On a velocity–time graph the gradient is a and the area under the line is s. The equations do not apply when the acceleration changes, for example once air resistance becomes significant.

Common mistakes

  • Using SUVAT when the acceleration is not constant: the equations only work for uniform acceleration.
  • Getting the signs wrong. Once you choose a positive direction, a deceleration or the acceleration due to gravity in the opposite direction must be negative.
  • Confusing displacement with distance: if an object turns back, s is its change in position, not the total distance travelled.
  • Forgetting the square root at the end when using v² = u² + 2as, or writing (u + 2as)² instead of u² + 2as.
  • Using velocities in km/h with an acceleration in m/s². Convert every velocity to m/s first.

Related formulas

AccelerationSpeed, distance and timeKinetic energyAll physics equations

Frequently asked questions

v = u + at, s = ut + ½at², s = vt − ½at², s = ½(u + v)t and v² = u² + 2as. They all assume constant acceleration in a straight line.

The five quantities in the equations: s displacement, u initial velocity, v final velocity, a acceleration and t time. In SI units they are measured in m, m/s, m/s² and s.

Write down the three values you know and the one you want. The fifth quantity is the one to leave out, so use the equation that does not contain it.

Yes. It is one of the equations given on the GCSE physics equation sheet, so you need to know how to apply it rather than memorise it. A-level physics data booklets list all the SUVAT equations, but you still need to choose the right one.

Yes, as long as air resistance is small enough to ignore. The acceleration is then g, about 9.8 m/s² downwards, so if you take upwards as positive, a = −9.8 m/s².