SUVAT Equations of Motion Calculator

Solve all 5 SUVAT equations of motion. Enter any 3 known values and find the remaining unknowns.

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SUVAT Equations Apply to Constant Acceleration

The SUVAT equations connect displacement, initial velocity, final velocity, acceleration, and time when acceleration is constant. The five symbols are commonly s, u, v, a, and t. Different equations eliminate different unknowns, so the best equation is the one containing the quantities you know and the one you need while excluding an unwanted variable.

Signs carry the direction information. If upward is positive, gravitational acceleration near Earth is approximately −9.81m/s2. A negative velocity then means motion downward, not a negative speed. The equations are exact for constant acceleration but do not apply directly when acceleration varies significantly with time or position.

v=u+at,   s=ut+½at2,   v2=u2+2as
SymbolMeaningWhy it appears / units
sDisplacementm; signed change in position, not total path length.
u, vInitial and final velocitym/s; signs depend on the chosen axis.
aConstant accelerationm/s².
tElapsed times; normally nonnegative for the interval considered.

Before substituting numbers, choose a positive direction and keep it consistent. A quadratic equation may produce more than one mathematical time; only roots consistent with the physical interval and conditions should be retained.

Constant acceleration links the SUVAT equations together. A correct solution from v=u+at should agree with displacement obtained from s=(u+v)t/2 when the same data apply. Conflicting results usually indicate a sign error, inconsistent time interval, or use of a constant-acceleration equation for a nonconstant process.

Worked Examples

Example 1: Car brakes: u=30m/s, v=0, a=-5m/s²
3 knowns: u,v,a
Result: t=6s, s=90m
Stopping distance and time
Example 2: Free fall: u=0, a=9.81, t=3s
3 knowns: u,a,t
Result: v=29.43m/s, s=44.1m
Dropped object after 3 seconds
Example 3: Car accelerating
u=5m/s, a=2m/s², t=4s
Result: v=13m/s, s=36m
Two SUVAT equations provide both final velocity and displacement.
Example 4: Ball thrown upward
u=19.62m/s, a=−9.81m/s², v=0
Result: t=2.00s to the top, s=19.62m
Setting final vertical velocity to zero identifies the highest point.

Common Mistakes

⚠️
Using distance where displacement is required

SUVAT uses signed displacement. A round trip can have nonzero distance but zero net displacement.

⚠️
Changing sign convention halfway through

Choose one positive direction first. Velocity, acceleration, and displacement signs must all follow that same axis.

⚠️
Using SUVAT for variable acceleration

If acceleration changes substantially, use calculus, numerical integration, or a model appropriate to the force law instead of constant-a equations.

Frequently Asked Questions

Which 3 to enter?
Any combination works: u,v,t → finds s,a. u,a,s → finds v,t. s,u,t → finds v,a. The calculator solves all 5 SUVAT equations simultaneously.
SUVAT only applies when?
Constant (uniform) acceleration only. For variable acceleration, calculus (integration of a(t)) is required. Gravity near Earth's surface is constant at 9.81 m/s².
How do I choose which SUVAT equation to use?
List the five quantities s,u,v,a,t, mark what is known, and choose an equation that includes the desired unknown but omits the quantity you do not know.
Can time be negative in SUVAT?
In a chosen mathematical coordinate description it can appear when extending a trajectory backward, but for an elapsed interval in a standard problem the physically relevant time is normally nonnegative.
Why can a quadratic SUVAT equation give two times?
A moving object can pass the same position at two different times, such as once while rising and again while falling. Both roots may be meaningful depending on the problem.
Does free fall always use −9.81 m/s²?
The magnitude near Earth’s surface is about 9.81m/s², but its sign depends on the chosen positive direction. The value also varies slightly with altitude and location.
When are the SUVAT equations valid?
The standard SUVAT equations assume constant acceleration over the time interval. They are not valid unchanged when acceleration depends strongly on time, position, or velocity. Choose signs for displacement, velocity, and acceleration consistently, then select an equation containing the known variables and the single unknown.

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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