What are the SUVAT equations?

SUVAT refers to the five variables in uniform-acceleration kinematics: s (displacement), u (initial velocity), v (final velocity), a (acceleration), and t (time). The four equations linking them are: v = u + at; s = ut + 0.5at2; v2 = u2 + 2as; and s = 0.5(u + v)t. Knowing any three variables lets you solve for the other two.

When can I use SUVAT equations?

SUVAT equations only apply when acceleration is constant (uniform) throughout the motion. Common cases include free-fall under gravity, a car braking uniformly, and a ball thrown vertically. If acceleration varies with time, you need calculus-based kinematics instead.

What is displacement in SUVAT?

Displacement (s) is the net change in position from start to finish, measured in metres. It is a vector quantity, so it can be positive or negative depending on direction. It differs from total distance travelled if the object reverses direction during the motion.

Kinematic Equations (SUVAT) Solver

This calculator solves the five kinematic equations for uniformly accelerated motion, commonly called the SUVAT equations. The five variables are s (displacement in metres), u (initial velocity in m/s), v (final velocity in m/s), a (acceleration in m/s squared), and t (time in seconds). If you know any three of these five variables, the equations can determine the remaining two. You tick the three variables you know, enter their values, and the calculator finds the two unknowns. The four underlying equations are v = u + at, s = ut + 0.5 at squared, v squared = u squared + 2as, and s = 0.5(u + v)t. Together they cover every combination of three known variables. This tool is particularly useful for students studying NCEA physics, A-level mechanics, or first-year university physics, as well as engineers analysing vehicle braking distances, athletes studying sprint acceleration, and anyone working through projectile or free-fall problems. The only requirement is that acceleration must be constant throughout the motion. If the object changes acceleration during the interval, such as a rocket burning fuel, these equations give only an approximation. Always confirm which three variables you genuinely know before entering values, as an incorrect set of knowns will produce a nonsensical result.

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u - Initial velocity

m/s

a - Acceleration

m/s²

t - Time

s - Displacement

v - Final velocity

m/s

44.15 m

displacement (s)

Final velocity (v)29.43 m/s

How it works

The calculator reads which three checkboxes are ticked to determine the combination of knowns. It then selects the pair of SUVAT equations that directly yields the two unknowns. The five equations used are: v = u + at; s = ut + ½at²; v² = u² + 2as; s = ½(u + v)t; and the derived s = vt - ½at². Where a quadratic arises, the calculator picks the physically positive root. Exactly three boxes must be ticked; ticking fewer or more triggers an error message.

Worked example

An object starts from rest (u = 0) under gravitational acceleration (a = 9.81 m/s²) for 3 seconds. From s = ut + ½at² = 0 + 0.5 × 9.81 × 9 = 44.15 m. Final velocity v = u + at = 0 + 9.81 × 3 = 29.43 m/s. Tick u, a, and t with the default values above to confirm these outputs.

Related calculators

Acceleration Calculator: quick solve for a = (v - u) / t with one unknown.
Free Fall Calculator: special case of SUVAT where u = 0 and a = 9.81 m/s2.
Projectile Motion Calculator: applies SUVAT in two dimensions for launched objects.
Velocity Calculator: constant-speed version using v = d / t.

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