Kinematics Calculator — SUVAT Equations
Solve all five SUVAT kinematic equations for displacement, initial velocity, final velocity, acceleration, and time. Enter any three known values and instantly solve for the remaining two.
Quick Presets
Enter any 3 values — leave 2 blank
0/3 filledWhat Is the Kinematics Calculator — SUVAT Equations?
The SUVAT kinematics calculator solves for any two unknown variables given the three known ones. It automatically identifies which of the five equations of motion applies, performs the algebra, and presents the full step-by-step working — making it ideal for physics homework, engineering problems, and real-world motion analysis.
- ›Solves for any two unknowns — fill in the three fields you know; leave the other two blank and the calculator determines both simultaneously.
- ›Automatic equation selection — the tool inspects which variables are known and picks the minimal set of SUVAT equations needed. No manual formula-hunting required.
- ›Full unit conversion — displacement in m, km, cm, ft, or mi; velocity in m/s, km/h, mph, or ft/s; acceleration in m/s², ft/s², or g; time in s, ms, min, or h. All converted to SI before solving.
- ›Step-by-step working — a collapsible panel shows the chosen equation, the substitution with your values, and every algebraic step to the result.
- ›Verification pass — where a second equation exists, the answer is cross-checked automatically and any discrepancy flagged as an input inconsistency.
- ›Real-world presets — one-click scenarios for free fall, car braking, projectile launch, and train deceleration populate the inputs instantly.
Formula
The five SUVAT equations describe uniformly accelerated motion. Given any three of the five variables, the remaining two can always be determined. This calculator selects the correct equation automatically.
Equation 1 — velocity–time
v = u + at
Uses: v, u, a, t (no s)
Equation 2 — displacement–time
s = ut + ½at²
Uses: s, u, a, t (no v)
Equation 3 — velocity–displacement
v² = u² + 2as
Uses: v, u, a, s (no t)
Equation 4 — average velocity
s = ½(u + v)t
Uses: s, u, v, t (no a)
Equation 5 — reverse-time form
s = vt − ½at²
Uses: s, v, a, t (no u)
| Symbol | Name | Description |
|---|---|---|
| s | Displacement | Net change in position from start to end point (metres by default) |
| u | Initial velocity | Velocity at the beginning of the time interval (m/s by default) |
| v | Final velocity | Velocity at the end of the time interval (m/s by default) |
| a | Acceleration | Constant rate of change of velocity (m/s² by default) |
| t | Time | Duration of the motion interval (seconds by default) |
How to Use
- 1Identify your three known variables: Decide which of s, u, v, a, t you already know. You need exactly three — any combination is accepted.
- 2Enter the known values: Type each known value into its field. Leave the two unknowns blank (or clear them). The calculator ignores empty fields.
- 3Select units (optional): Use the unit dropdowns beside each field to work in km/h, mph, feet, g-units, or any other supported unit. Conversion to SI is automatic.
- 4Press Calculate or Enter: Click the orange Calculate button or press Enter. Results appear instantly in two result cards — one for each solved variable.
- 5Review the working: Expand the Step-by-Step panel to see exactly which SUVAT equation was chosen, how your values were substituted, and each algebraic step. A verification check confirms consistency.
Example Calculation
Example 1 — Car accelerating from rest to motorway speed
A car starts from rest and reaches 60 mph (26.82 m/s) in 8 seconds. Find the acceleration and distance covered.
Known: u = 0 m/s, v = 26.82 m/s, t = 8 s
Equation chosen: v = u + at (no s needed)
a = (v − u) / t = (26.82 − 0) / 8
a = 3.35 m/s² (≈ 0.34 g)
Equation chosen: s = ½(u + v)t (no a needed)
s = 0.5 × (0 + 26.82) × 8
s = 107.3 m
Example 2 — Ball in free fall from 50 m
A ball is dropped from rest at a height of 50 m. Find the time to reach the ground and the impact velocity (taking downward as positive, g = 9.81 m/s²).
Known: u = 0 m/s, a = 9.81 m/s², s = 50 m
Equation chosen: s = ut + ½at² (no v needed)
50 = 0 × t + 0.5 × 9.81 × t²
t² = 100 / 9.81 = 10.194
t = √10.194 ≈ 3.19 s
Equation chosen: v² = u² + 2as (no t needed)
v² = 0 + 2 × 9.81 × 50 = 981
v = √981 ≈ 31.32 m/s on impact
Verification
Understanding Kinematics — SUVAT Equations
The Five Equations of Motion
The SUVAT equations — named from their five variables s, u, v, a, t — were developed by Galileo Galilei in the early 1600s and later formalised in the notation we use today. They apply to any object moving with constant (uniform) acceleration along a straight line. Each of the five equations omits one variable, which is why there are exactly five: one for each variable left out.
- ›v = u + at — The most fundamental. Velocity changes linearly with time when acceleration is constant. Rearranges to find a = (v − u)/t (the definition of uniform acceleration).
- ›s = ut + ½at² — Displacement as a quadratic function of time. The ½at² term is the extra distance gained from the increasing velocity.
- ›v² = u² + 2as — The energy-equivalent form. Derived by eliminating t; useful when time is unknown. Forms the basis for stopping-distance calculations.
- ›s = ½(u + v)t — Uses the average velocity. Elegant when both u and v are known but a is not. Distance = average speed × time.
- ›s = vt − ½at² — The reverse-time form; it uses final velocity v instead of initial u. Useful for problems stated backwards in time.
Choosing the Right SUVAT Equation
The choice of equation depends purely on which variable is absent from your problem. If you have three knowns, identify which variable is missing from the list {s, u, v, a, t} and use the equation that does not contain that variable:
| Variable absent | Equation to use | Typical scenario |
|---|---|---|
| s (displacement) | v = u + at | Finding speed after braking for t seconds |
| u (initial velocity) | v² = u² + 2as → rearranged | Estimating entry speed from skid marks |
| v (final velocity) | s = ut + ½at² | Distance a rocket travels in first 10 s of burn |
| a (acceleration) | s = ½(u + v)t | Average-velocity problems — no force info needed |
| t (time) | v² = u² + 2as | Stopping distance from v to 0 over measured s |
When three knowns are supplied the calculator performs this logic automatically: it inspects which two variables are blank, finds the intersection of equations that contain the knowns, and solves the simplest one first. The second unknown is then found either from a second SUVAT equation or by direct substitution back into the first.
Sign Convention for Displacement
SUVAT is a vector framework. Positive and negative signs carry physical meaning:
- ›Choose a positive direction at the start (e.g. rightward or upward) and stick to it throughout the problem.
- ›A negative displacement s means the object ended up behind its start position.
- ›A negative velocity means the object is moving in the negative direction (e.g. downward if up is positive).
- ›A negative acceleration means the object is decelerating if it moves in the positive direction, or accelerating in the negative direction.
- ›Free-fall problems: if you define downward as positive, a = +9.81 m/s². If upward is positive, a = −9.81 m/s².
Common sign-convention mistake
When SUVAT Doesn't Apply
The equations are valid only under uniform (constant) acceleration. They break down in several common scenarios:
- ›Variable acceleration — e.g. a rocket burning fuel (thrust decreases as mass decreases). Use calculus: v = ∫a dt, s = ∫v dt.
- ›Circular motion — constant speed in a circle involves centripetal acceleration changing direction; SUVAT applies only to the tangential component.
- ›Relativistic speeds — above ~10% of the speed of light (≈ 3×10⁷ m/s), special-relativistic corrections are needed. SUVAT overestimates displacement.
- ›Air resistance — drag creates velocity-dependent deceleration. Free-fall with drag requires differential equations, not SUVAT.
Real-World Kinematics
| Application | Typical SUVAT problem | Key equation |
|---|---|---|
| Automotive safety | Minimum braking distance at highway speed | v² = u² + 2as (v=0) |
| Sports science | Acceleration of a sprinter off the blocks | v = u + at |
| Aerospace | Runway length needed for take-off | s = ½(u + v)t |
| Ballistics | Muzzle velocity from barrel length and shot time | v² = u² + 2as |
| Civil engineering | Train stopping distance for signal spacing | v² = u² + 2as |
| Space launch | Altitude reached in first burn phase | s = ut + ½at² |
Frequently Asked Questions
What does SUVAT stand for?
SUVAT stands for the five variables in the equations of motion:
- ›s — displacement (net change in position, not total distance travelled)
- ›u — initial velocity (speed and direction at the start of the interval)
- ›v — final velocity (speed and direction at the end of the interval)
- ›a — acceleration (constant rate of velocity change)
- ›t — time (duration of the motion interval)
The equations were derived by Galileo in the early 17th century and underpinned Newton's mechanics. They remain the standard approach for constant-acceleration problems across physics, engineering, and sports science.
How do I choose which SUVAT equation to use?
Identify the variable that appears in neither your knowns nor your unknowns, then use the equation that omits it:
- ›Missing s → use v = u + at
- ›Missing u → use s = vt − ½at² or v² = u² + 2as rearranged
- ›Missing v → use s = ut + ½at²
- ›Missing a → use s = ½(u + v)t
- ›Missing t → use v² = u² + 2as
This calculator automates the selection — just enter three values and leave two blank. It identifies the missing variable pattern and picks the optimal equation.
What does negative acceleration mean in SUVAT?
Negative acceleration simply means the acceleration vector points opposite to your chosen positive direction:
- ›If rightward is positive and a car brakes, a is negative (it decelerates).
- ›If upward is positive and a ball is in free fall, a = −9.81 m/s².
- ›"Deceleration" is not a separate concept — it just means a and v have opposite signs.
- ›A negative a does not always mean slowing down: if v is also negative (moving in the negative direction), a negative a actually speeds the object up.
Key insight
Speed increases when a and v have the same sign. Speed decreases when they have opposite signs. This calculator shows a vector note whenever you enter a negative value.
Can SUVAT be applied to horizontal and vertical projectile motion separately?
Projectile motion under gravity (no air resistance) splits into two independent 1-D problems:
- ›Horizontal axis: a = 0. SUVAT gives s_x = u_x × t (constant velocity).
- ›Vertical axis: a = −g = −9.81 m/s². Apply all five SUVAT equations normally.
- ›Time t is the same for both axes — solve the vertical problem for t first, then substitute into the horizontal equation.
This kinematics calculator handles one axis at a time. For the full 2-D trajectory (range, max height, time of flight) use the Projectile Motion Calculator.
Do SUVAT equations work at relativistic speeds?
SUVAT is a classical-mechanics tool and is accurate only at speeds well below the speed of light:
- ›At 1% of c (3×10⁶ m/s), the relativistic correction is only 0.005% — SUVAT is effectively exact.
- ›At 10% of c, the Lorentz factor γ ≈ 1.005 — a 0.5% error, often acceptable.
- ›At 50% of c, γ ≈ 1.155 — SUVAT overestimates displacement by 15.5%.
- ›The fastest human-made objects (Voyager 1: ~17 km/s = 0.006% c) are well within the SUVAT regime.
How does free fall work in SUVAT?
Free fall is the simplest constant-acceleration scenario: a = g throughout.
- ›Dropped from rest (u = 0), downward positive: s = ½ × 9.81 × t², v = 9.81t, v² = 2 × 9.81 × s.
- ›Thrown upward (upward positive): u is positive, a = −9.81. At max height, v = 0, so s_max = u²/(2 × 9.81).
- ›Time of flight for a projectile thrown up with speed u: t_total = 2u/9.81 (returns to same height).
- ›Impact velocity after falling height h: v = √(2gh) regardless of mass (Galileo's experiment).
This calculator includes a Free Fall preset (u = 0, a = 9.81 m/s²) so you can enter just the height or time and solve for the other instantly.
Why do three known values always determine the other two in SUVAT?
The five SUVAT equations are not independent — they all derive from two basic integrals:
v = u + at (integrate: a = dv/dt with constant a)
s = ut + ½at² (integrate again: v = ds/dt)
The other three equations are algebraic rearrangements of these two.
Since there are only two fundamental equations, specifying three of the five variables provides enough constraints to solve the remaining two. However:
- ›If you supply four or five values, the system is over-determined and may be inconsistent.
- ›Inconsistency (e.g. v = u + at and s ≠ ½(u+v)t simultaneously) signals a measurement error or a non-constant acceleration.
- ›This calculator detects over-determined inputs and flags them as inconsistent.