The Five Equations of Uniform Acceleration
When an object accelerates at a constant rate, its motion is entirely described by five variables: displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t). From these five SUVAT variables, five independent equations emerge.
Each equation omits one variable, making it useful when you know three others and need the fourth. For instance, if time is unknown or irrelevant, you can relate velocity change directly to displacement. Conversely, if displacement isn't measured, you can connect velocity change to time. The five equations are not separate discoveries—they're rearrangements of the same underlying physics.
These equations apply to:
- Free-fall under gravity
- Vehicle acceleration or braking
- Projectile motion
- Any scenario with constant acceleration
The SUVAT Equations Explained
The core equations derive from the definitions of acceleration and average velocity. Below are the five fundamental rearrangements:
v = u + at
s = (u + v) ÷ 2 × t
v² = u² + 2as
s = ut + 0.5at²
s = vt − 0.5at²
s— Displacement (distance travelled from starting position, in metres)u— Initial velocity (velocity at the start of the motion, in m/s)v— Final velocity (velocity at the end of the motion, in m/s)a— Acceleration (constant rate of change of velocity, in m/s²)t— Time (duration of the motion, in seconds)
Understanding Displacement vs. Distance
A critical distinction: displacement measures position change from start to finish, while distance is the total path length travelled. If an object returns to its starting point, displacement is zero, but distance is not.
On a velocity–time graph, displacement is represented by the area under the curve. For constant acceleration, this area forms a trapezoid. The parallel sides are the initial and final velocities, and the height is time. Averaging the two velocities and multiplying by time gives displacement:
s = (u + v) ÷ 2 × t
This geometric relationship is why the kinematic equations work. Every rearrangement traces back to this visual truth on the v–t graph.
Solving Without Known Time
Sometimes you have displacement, initial velocity, and final velocity—but time isn't given or measured. Rather than find time first, you can eliminate it algebraically. Starting with v = u + at, rearrange to get t = (v − u) ÷ a. Substitute this into s = (u + v) ÷ 2 × t:
s = (u + v) ÷ 2 × (v − u) ÷ a
Simplify to obtain:
v² = u² + 2as
This form is invaluable for collisions, stopping distances, and any scenario where time is unmeasured. Similarly, if you know time but not displacement, you can combine equations to find acceleration or velocity without needing s.
Common Pitfalls and Practical Notes
Avoid these frequent mistakes when applying SUVAT equations:
- Watch your signs — Acceleration and velocity are vectors. Upward motion with downward gravity means <em>a</em> is negative. A car slowing down has negative acceleration. Consistent sign convention prevents wrong answers.
- Distinguish between starting from rest and starting with velocity — Setting u = 0 simplifies equations dramatically but is only valid if the object begins stationary. A vehicle at 10 m/s that then accelerates has u = 10, not 0.
- Check unit consistency — Mixing metres with kilometres or seconds with hours will corrupt the result. Convert everything to SI units (metres, seconds, m/s, m/s²) before entering the calculator.
- Recall that these equations assume constant acceleration — Real-world friction, air resistance, and engine torque curves mean acceleration varies. SUVAT is an approximation valid over short intervals or in controlled conditions like free-fall in a vacuum.