Constant-acceleration motion
What are kinematic equations?
Kinematic equations describe motion when acceleration remains constant. They connect displacement, initial velocity, final velocity, acceleration, and time.
These equations are commonly called SUVAT equations because each letter represents one motion variable.
SUVAT variables
Meaning of each variable
- s is displacement in metres.
- u is initial velocity in metres per second.
- v is final velocity in metres per second.
- a is acceleration in metres per second squared.
- t is elapsed time in seconds.
Formula reference
Four kinematic equations
Velocity–timev = u + at
Displacement–times = ut + ½at²
Velocity–displacementv² = u² + 2as
Average-velocity forms = ½(u + v)t
Equation selection
How to choose the correct equation
Select an equation that contains the variable you want to calculate and the values you already know.
- Use v = u + at when displacement is not required.
- Use s = ut + ½at² when final velocity is not known.
- Use v² = u² + 2as when time is not known.
- Use s = ½(u + v)t when acceleration is not required.
Worked example
Calculate final velocity
An object has an initial velocity of 5 m/s, acceleration of 2 m/s², and moves for 4 seconds.
- Choose v = u + at.
- Substitute v = 5 + (2 × 4).
- Simplify v = 5 + 8.
- Final velocity is 13 m/s.
Worked example
Calculate displacement
An object starts at 5 m/s, accelerates at 2 m/s², and moves for 4 seconds.
- Choose s = ut + ½at².
- Substitute s = (5 × 4) + ½(2)(4²).
- Calculate s = 20 + 16.
- Displacement is 36 m.
Direction and signs
Positive and negative values
Velocity, acceleration, and displacement can be positive or negative depending on the selected direction. Use one direction as positive and keep that sign convention throughout the calculation.
Time must remain positive because it represents elapsed duration.
Validity
When not to use SUVAT equations
These equations assume constant acceleration. They should not be used directly when acceleration changes continuously with time or position.
For variable acceleration, calculus, numerical methods, or motion graphs may be required.
Related tools
Continue your motion analysis
Use the Acceleration Calculator for direct velocity-change calculations.
Use the Displacement Calculator to compare initial and final positions.
Use the Average Velocity Calculator to calculate displacement per unit time.
Use the Free Fall Calculator for vertical motion under gravity.
Questions and answers
Kinematic equations FAQ
What are the SUVAT variables?
SUVAT represents displacement, initial velocity, final velocity, acceleration, and time. These variables describe motion with constant acceleration.
Which kinematic equations does this calculator use?
The calculator uses v = u + at, s = ut + ½at², v² = u² + 2as, and s = ½(u + v)t.
Can the calculator solve for any SUVAT variable?
It can solve variables supported by the selected equation. Choose an equation that contains the unknown variable and enter the remaining known values.
Can acceleration be negative?
Yes. Negative acceleration can represent deceleration or acceleration in the selected negative direction.
Why might a velocity equation have no real solution?
The values may produce a negative quantity under a square root in v² = u² + 2as. In that case, no real-valued velocity satisfies the supplied data.
When are SUVAT equations valid?
SUVAT equations are valid when acceleration is constant during the motion interval.