Kinematic Equations (SUVAT) Solver

This calculator solves the five equations of motion, known as SUVAT, for any problem involving constant (uniform) acceleration. SUVAT stands for the five variables describing straight-line motion: displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t). Because the five equations link these variables together, you only need to know three of them to work out the other two exactly, without guessing which formula applies. You enter any three of the five values, in metres, metres per second, metres per second squared and seconds, and leave the two unknowns blank. The solver picks the correct equation automatically, fills in every result across all five variables, shows which equation or equations were applied, and produces a step-by-step worked solution showing how each figure was reached. Negative numbers are accepted and expected, since deceleration, downward motion under gravity, and reverse displacement are all represented with a minus sign once you have chosen a consistent positive direction. Five quick-fill presets, including free fall from rest, a car braking to a stop, a ball thrown upward, constant velocity motion, and a plane reaching takeoff speed, let you see worked scenarios instantly. The tool suits NCEA, A-Level and first-year university physics, and it only applies where acceleration stays constant throughout the motion; it is not suitable for changing acceleration, circular motion, or two-dimensional projectile motion.

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Standard Physics Classical Newtonian mechanics. Equations apply to constant (uniform) acceleration only.

Enter exactly 3 values to solve (0 entered so far)

1. Known Values

s - Displacement (leave blank to solve)

u - Initial velocity (leave blank to solve)

m/s

v - Final velocity (leave blank to solve)

m/s

a - Acceleration (leave blank to solve)

m/s²

t - Time (leave blank to solve)

2. Quick-Fill Example

Load a preset scenario to see how the solver works.

Scenario

All Five Variables

Displacement

metres

Initial velocity

m/s

Final velocity

m/s

Acceleration

m/s²

Time

seconds

Worked Solution

Enter three values above to see the step-by-step working.

The Five Kinematic (SUVAT) Equations

The five equations of motion describe the relationship between displacement, velocity, acceleration, and time for an object moving with constant acceleration. They were derived from the definitions of velocity and acceleration and are the foundation of classical mechanics at NCEA, A-Level, and first-year university physics.

Equation	Form	Variable not included
1	v = u + at	s (displacement)
2	s = ut + ½at²	v (final velocity)
3	v² = u² + 2as	t (time)
4	s = ½(u + v)t	a (acceleration)
5	s = vt − ½at²	u (initial velocity)

Each equation omits one variable, so once you know three variables you can always find an equation that contains exactly one unknown and solve it directly. When only three values are known and none of the five single-variable forms applies directly (rare in introductory problems), the solver above uses algebraic substitution between equations.

Worked Example: Default Values

A car travelling at u = 20 m/s brakes with a = -5 m/s² for t = 4 s. Find v and s.

Step 1: Find v using v = u + at
v = 20 + (-5)(4) = 20 - 20 = 0 m/s

Step 2: Find s using s = ut + ½at²
s = 20(4) + ½(-5)(4²) = 80 + ½(-5)(16) = 80 - 40 = 40 m

The car decelerates to rest in 4 seconds, covering 40 metres. This matches the braking scenario preset above.

Sign Convention

You must choose a positive direction and apply it consistently. Common choices:

Horizontal motion: right is positive, left is negative.
Vertical motion: upward is positive, downward is negative (so g = -9.81 m/s²).
Some textbooks define downward as positive when studying falling objects, making g = +9.81 m/s². Either convention works as long as it is consistent throughout the problem.

When SUVAT Does Not Apply

The SUVAT equations require acceleration to be constant throughout the motion. They do not apply when:

Acceleration varies with time (for example, a rocket burning fuel, air resistance proportional to velocity).
The object moves in a circle (acceleration is centripetal and changes direction continuously).
The object experiences a varying force (for example, a spring system in simple harmonic motion).

For non-constant acceleration, you need calculus: integration of the acceleration function to find velocity, and integration again to find displacement.

Common NZ Physics Curriculum Values

Constant	Value	Context
Gravitational acceleration (g)	9.81 m/s² (downward)	Free fall, projectile motion near Earth's surface
Speed of sound in air	~343 m/s at 20°C	Wave motion (not a SUVAT problem)
Speed of light	3 × 10&sup8; m/s	Special relativity (not classical mechanics)

Related Calculators

Science and Engineering Calculators: full index of physics and engineering tools.
Acceleration Calculator: find acceleration from velocity change and time.
Average Velocity Calculator: displacement divided by time.
Angular Velocity Calculator: rotational motion.
Projectile Motion Calculator: two-dimensional motion under gravity.

Sources and method: Equations of motion derived from definitions a = dv/dt and v = ds/dt with constant acceleration. Standard form as used in NCEA Physics (Achieve/Merit/Excellence) and international A-Level syllabi. Values of g = 9.81 m/s² per NIST/BIPM conventions.

This solver applies the five standard kinematic equations and assumes constant (uniform) acceleration throughout the motion. For problems with varying acceleration, projectile motion (two dimensions), or relativistic speeds, additional tools are required. Always verify the sign convention you have chosen is consistent across all variables entered.

Frequently asked questions

What are the five SUVAT equations?

The five kinematic equations of motion (SUVAT) are: (1) v = u + at, (2) s = ut + (1/2)at squared, (3) v squared = u squared + 2as, (4) s = (1/2)(u + v)t, and (5) s = vt - (1/2)at squared. Each equation links four of the five variables: s (displacement), u (initial velocity), v (final velocity), a (acceleration), and t (time). Given any three variables, you can always choose an equation that does not contain the fourth unknown variable and solve directly.

When can SUVAT equations be used?

SUVAT equations apply only when acceleration is constant (uniform acceleration) throughout the motion. They are used in classical mechanics for problems such as free fall under gravity (a = 9.81 m/s squared downward), a car braking at a steady rate, or a ball rolled along a frictionless surface. They do not apply when acceleration is changing (for example, a rocket burning fuel, or circular motion where direction is continuously changing).

What sign convention should I use for SUVAT problems?

Choose a positive direction at the start and keep it consistent throughout the problem. For example, if you define upward as positive, then a ball thrown upward has positive initial velocity, acceleration due to gravity is negative (-9.81 m/s squared), and displacement above the starting point is positive while displacement below is negative. Entering a negative value in the solver correctly handles deceleration, downward motion, or reverse displacement.

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