SUVAT Calculator — Solve Any Kinematic Equation Instantly
Enter any three of the five kinematic variables — displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t) — and this calculator solves for the remaining two using the correct SUVAT equation, with full step-by-step substitution shown.
SUVAT Calculator Tool
Fill in exactly 3 of the 5 fields below and leave the other 2 blank — the calculator detects which values you've given and solves for the rest automatically.
Quick-load a scenario:
How This Calculator Works
The five SUVAT equations describe the motion of any object moving in a straight line with constant (uniform) acceleration. "SUVAT" is a mnemonic for the five variables: displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).
You enter any three known values, and the calculator automatically selects the correct equation — or pair of equations — to solve for the remaining two unknowns. Every result shows the equation used, the substitution of your values, and the algebraic steps to the final answer.
All calculations assume constant acceleration in one dimension. This model applies to free fall under gravity, vehicles braking or accelerating on straight roads, objects sliding down frictionless inclined planes, and any motion where the net force (and therefore acceleration) remains constant throughout the interval.
The Five SUVAT Equations
Each equation connects four of the five variables, omitting one. The calculator selects the equation that uses your three known values plus the unknown you need.
Equation 1 (no s)
Final velocity equals initial velocity plus acceleration multiplied by time.
Equation 2 (no v)
Displacement equals initial velocity multiplied by time plus half the acceleration multiplied by time squared.
Equation 3 (no u)
Displacement equals final velocity multiplied by time minus half the acceleration multiplied by time squared.
Equation 4 (no t)
Final velocity squared equals initial velocity squared plus twice the acceleration multiplied by displacement.
Equation 5 (no a)
Displacement equals the average of initial and final velocities multiplied by time.
These five equations are not independent — any one can be derived from two others. But having all five available lets the calculator solve for any pair of unknowns directly, without needing intermediate steps.
Units Supported
The calculator accepts inputs in both metric and imperial systems with per-field unit selection.
Displacement (s)
Metres (m), kilometres (km), centimetres (cm), feet (ft), miles (mi). A 100m sprint track is 100m. A typical car braking distance at 100 km/h is 56m on dry road (based on UK Highway Code data). A commercial aircraft takeoff roll on a Boeing 737 is approximately 1,500–2,000m.
Initial Velocity (u) and Final Velocity (v)
Metres per second (m/s), kilometres per hour (km/h), feet per second (ft/s), miles per hour (mph). Walking speed is approximately 1.4 m/s (5 km/h). The world 100m sprint record (9.58 seconds by Usain Bolt, 2009) corresponds to an average speed of 10.44 m/s (37.6 km/h) with a peak of approximately 12.4 m/s (44.7 km/h). The speed of sound at sea level is 343 m/s (1,235 km/h).
Acceleration (a)
m/s², km/h/s, ft/s², g-force (where 1g = 9.80665 m/s²). Gravity is 9.81 m/s² downward. A Formula 1 car accelerates at approximately 15 m/s² (1.5g) during launch. Emergency braking in a modern passenger car produces deceleration of 8–10 m/s² (0.8–1.0g) on dry asphalt. Fighter jet pilots experience up to 9g (88.3 m/s²) during tight turns.
Time (t)
Seconds (s), milliseconds (ms), minutes (min). A car airbag deploys in approximately 30 milliseconds. A dropped object takes 0.45 seconds to fall 1 metre. A bullet from an AK-47 takes approximately 0.37 seconds to travel 300m.
Quick-Load Presets
Pre-configured scenarios load realistic values for instant exploration. Each preset uses measured or estimated values from real situations.
Solve Mode — Choose Your Three Known Variables
Unlike simpler calculators that require you to pick a specific equation, this calculator detects which three fields you've filled and automatically selects the correct SUVAT equation. There are exactly 10 possible combinations of three knowns from five variables:
| Known | Equations Used |
|---|---|
| s, u, v | Equation 4 (v² = u² + 2as) and Equation 5 (s = ½(u+v)t) |
| s, u, a | Equation 2 (s = ut + ½at²) and Equation 1 (v = u + at) |
| s, u, t | Equation 5, then derives a from Equation 2 |
| s, v, a | Equation 3 and Equation 4 |
| s, v, t | Equation 5, then derives a from Equation 3 |
| s, a, t | Equation 2 and Equation 1 |
| u, v, a | Equation 4 and Equation 1 |
| u, v, t | Equation 1 and Equation 5 |
| u, a, t | Equation 1 and Equation 2 |
| v, a, t | Equation 1 and Equation 3 |
The calculator handles all 10 combinations. You never need to select an equation yourself.
SUVAT Calculator for Free Fall and Gravity
Free fall is the simplest application of SUVAT equations. The initial velocity is zero (for a dropped object), acceleration equals g (9.81 m/s² downward), and air resistance is neglected.
How Fast Does a Dropped Object Hit the Ground?
From height h, using v² = u² + 2as with u = 0 and a = g: v = √(2gh). A ball dropped from 20m hits the ground at v = √(2 × 9.81 × 20) = 19.8 m/s (71.3 km/h). The calculator shows this step by step.
How Long Does It Take to Fall?
Using s = ½gt²: t = √(2s/g). From 20m: t = √(2 × 20 / 9.81) = 2.02 seconds.
Real-world context: Galileo reportedly dropped objects from the Leaning Tower of Pisa (height 56m) in 1589 to demonstrate that all objects fall at the same rate regardless of mass. From 56m, a dropped object takes 3.38 seconds to reach the ground and arrives at 33.1 m/s (119 km/h). Felix Baumgartner's 2012 freefall from 39 km altitude reached a maximum speed of 373 m/s (1,342 km/h, Mach 1.25) — though this involved non-constant acceleration because air density varies with altitude.
SUVAT Calculator for Vehicle Braking and Road Safety
Braking distance calculations are one of the most important real-world applications of SUVAT equations. Using v² = u² + 2as with v = 0 (vehicle stops) gives the braking distance s = u² / (2|a|).
Braking Distances at Typical Decelerations
- Dry asphalt, good tyres: deceleration approximately 8–10 m/s² (0.8–1.0g)
- Wet asphalt: deceleration approximately 5–7 m/s² (0.5–0.7g)
- Icy road: deceleration approximately 1–2 m/s² (0.1–0.2g)
- Gravel or loose surface: deceleration approximately 3–5 m/s² (0.3–0.5g)
At 50 km/h (13.9 m/s) on dry road (a = −9 m/s²): braking distance = 13.9² / (2 × 9) = 10.7m. Time to stop = 1.54s.
At 100 km/h (27.8 m/s) on dry road: braking distance = 27.8² / (2 × 9) = 42.9m. Time to stop = 3.09s.
At 100 km/h on wet road (a = −6 m/s²): braking distance = 27.8² / (2 × 6) = 64.4m. Time to stop = 4.63s. That's 50% farther than dry road.
At 100 km/h on ice (a = −1.5 m/s²): braking distance = 27.8² / (2 × 1.5) = 257m. Time to stop = 18.5s. Over a quarter kilometre.
Doubling the speed quadruples the braking distance — a direct consequence of the v² term in the SUVAT equation. This is why speed limits exist and why the difference between 50 km/h and 60 km/h is far more dangerous than the numbers suggest: braking distance increases by 44%, not 20%.
The UK Highway Code, referenced in over 38 million driving theory tests per year, bases its stopping distance tables directly on SUVAT calculations. Total stopping distance = thinking distance (reaction time × speed) + braking distance (SUVAT).
SUVAT Calculator for Sports Performance
SUVAT equations quantify athletic performance in any sport involving acceleration along a straight line.
Sprinting
Usain Bolt's 100m world record (9.58s, Berlin 2009) can be modelled as constant acceleration followed by a constant-speed phase. His first 40m took approximately 4.64 seconds, giving an average acceleration of 3.72 m/s². His peak speed of 12.4 m/s (44.7 km/h) was reached at approximately 65m. For a simplified constant-acceleration model over the full 100m, the calculator gives a = 2.18 m/s² and a final velocity of 20.9 m/s — the latter is an overestimate because real sprinters decelerate slightly in the final 20m.
Cycling Acceleration
A track cyclist accelerating from a standing start in the velodrome reaches 60 km/h (16.7 m/s) in approximately 10 seconds, giving a = 1.67 m/s² and covering s = ½ × 1.67 × 10² = 83.3m during acceleration.
Swimming Starts
An Olympic swimmer leaves the starting block at approximately 4.5 m/s, decelerates in the water at roughly −1.5 m/s² to a cruising speed of 2.0 m/s. Time to reach cruising speed: t = (2.0 − 4.5) / (−1.5) = 1.67s. Distance covered during deceleration: s = 4.5(1.67) + ½(−1.5)(1.67²) = 5.42m — approximately one-quarter of a pool length.
Cricket Bowling
A fast bowler releases the ball at approximately 42 m/s (150 km/h). The ball decelerates through air resistance at roughly 2 m/s² over the 20.12m pitch length, arriving at the batsman at v = √(42² − 2 × 2 × 20.12) = 41.0 m/s. Time of flight = 0.49 seconds — giving the batsman less than half a second to react.
SUVAT Calculator for Engineering and Design
SUVAT equations are fundamental to mechanical engineering wherever constant acceleration or deceleration is designed into a system.
Elevator Design
Passenger comfort limits elevator acceleration to approximately 1.0–1.5 m/s² (0.1–0.15g). An elevator accelerating from rest to 6 m/s at 1.2 m/s² takes t = 6/1.2 = 5.0 seconds and covers s = ½ × 1.2 × 5² = 15m during acceleration. For a 200m-tall building, the elevator spends approximately 30m accelerating and decelerating (15m each) and the remaining 170m at constant speed, taking 170/6 = 28.3 seconds. Total trip time ≈ 38.3 seconds.
Conveyor Belt Startup
A 50m conveyor belt accelerating to operating speed of 2 m/s at 0.5 m/s² takes 4 seconds and covers 4m during acceleration. The remaining 46m runs at constant speed.
CNC Machine Tool Positioning
High-precision CNC routers accelerate tool heads at 5–20 m/s² (0.5–2g) to reach traverse speeds of 30–60 m/min (0.5–1.0 m/s). At 10 m/s² accelerating to 1 m/s, the acceleration distance is only 0.05m (50mm) — small enough that most of a 1m traverse is at constant speed.
Roller Coaster Design
A hydraulic launch coaster (like Kingda Ka at Six Flags) accelerates riders from 0 to 57 m/s (206 km/h) in 3.5 seconds. Using the SUVAT calculator: a = 57/3.5 = 16.3 m/s² (1.66g), and the launch track length is s = ½ × 16.3 × 3.5² = 99.8m — approximately 100m. The 2.0g threshold is generally considered the comfort limit for sustained forward acceleration on amusement rides.
SUVAT Calculator for Physics Coursework and Exams
SUVAT equations appear in virtually every introductory physics curriculum worldwide. They are tested in GCSE Physics and A-Level Physics in the UK, AP Physics 1 and AP Physics C in the United States, IB Physics SL and HL, and equivalent courses in every country.
Common exam question types this calculator solves:
- "A car accelerates from 10 m/s to 30 m/s over a distance of 200m. Find the acceleration and time taken." — Enter u = 10, v = 30, s = 200. Calculator uses v² = u² + 2as to find a = 2.0 m/s², then v = u + at to find t = 10s.
- "A ball is thrown vertically upward at 15 m/s. Find the maximum height and time to reach it." — Enter u = 15, v = 0 (at the peak), a = −9.81. Calculator uses v² = u² + 2as to find s = 11.47m, then v = u + at to find t = 1.53s.
- "A train decelerates at 0.5 m/s² from 25 m/s. How far does it travel before stopping?" — Enter u = 25, v = 0, a = −0.5. Calculator gives s = 625m and t = 50s.
The step-by-step solution feature shows the exact working a student would write in an exam, including formula selection, substitution, and algebraic rearrangement — useful for checking homework and understanding the method, not just the answer.
Direction Convention and Sign Handling
SUVAT equations are one-dimensional, so direction is encoded using positive and negative signs. The calculator follows the standard physics convention:
Positive direction: forward, upward, or rightward (your choice — as long as it's consistent within one problem).
Negative values mean the opposite direction. A negative acceleration means deceleration (if the object moves in the positive direction). A negative displacement means the object has moved backward from its starting point. A negative velocity means the object is moving in the negative direction.
Example — Ball Thrown Upward
Taking upward as positive, u = +20 m/s (thrown up), a = −9.81 m/s² (gravity pulls down). At the peak, v = 0. On the way back down, v becomes negative. The calculator handles all these sign conventions correctly.
Common sign errors to avoid: entering g as +9.81 when the object moves upward gives incorrect results. For upward motion, g must be entered as −9.81 (opposing the direction of motion). This is the single most common mistake students make with SUVAT problems.
When SUVAT Equations Do Not Apply
SUVAT equations require constant acceleration throughout the motion interval. They break down when acceleration varies with time, position, or velocity.
Variable Acceleration Scenarios
A car accelerating through its gears (engine torque varies), a rocket burning fuel (mass decreases, so acceleration increases even at constant thrust), a skydiver in freefall (air resistance increases with speed until terminal velocity), and oscillating systems like pendulums and springs (acceleration depends on position).
Circular Motion
Objects moving in circles have constantly changing direction of acceleration (centripetal acceleration points toward the centre). SUVAT equations do not apply to circular paths.
Multi-Stage Problems
A car accelerating, then cruising, then braking requires three separate SUVAT calculations — one for each constant-acceleration phase — with the final conditions of each phase becoming the initial conditions of the next.
For variable acceleration problems, calculus-based kinematics (integrating acceleration to find velocity and position) replaces the algebraic SUVAT approach. For a full exploration of when and why these equations work, see our SUVAT Equations — Complete Guide.
Accuracy and Precision
This calculator computes all results to full floating-point precision and displays up to 4 significant figures. No rounding is applied during intermediate steps. The quadratic formula is used to solve Equations 2 and 3 for time, with the calculator selecting the physically meaningful (positive) root.
All inputs accept positive and negative numbers. Negative displacement, negative velocity, and negative acceleration are all physically meaningful and are handled correctly according to the sign convention described above.
Quadratic Solutions
When solving for time using s = ut + ½at², the quadratic formula yields two roots. The calculator displays the positive root by default. In cases where both roots are positive (for example, a ball thrown upward passes through a certain height twice — once going up and once coming down), the calculator displays both solutions and labels them accordingly.
Related Calculators and Tools
Calculators
Articles
Related Simulation
Frequently Asked Questions
What is a SUVAT calculator?
A SUVAT calculator solves the five standard kinematic equations for uniformly accelerated straight-line motion. You enter any three of the five variables — displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t) — and the calculator solves for the remaining two. It automatically selects the correct equation based on which variables you provide.
What does SUVAT stand for?
SUVAT is a mnemonic for the five kinematic variables: s (displacement), u (initial velocity), v (final velocity), a (acceleration), and t (time). The letters come from standard physics notation, with u and v deriving from the Latin terms for initial and final states. The term is used primarily in UK, Australian, and South Asian physics education.
How many SUVAT equations are there?
There are five standard SUVAT equations, each connecting four of the five variables. Any one equation can be derived from two others, so only three are truly independent. However, having all five available means you can always solve a problem in a single step without needing to find intermediate unknowns first.
Which SUVAT equation should I use?
Identify the variable you do not know and do not need. Choose the equation that omits that variable. For example, if you know u, a, and t, and want to find s, you don't need v — use s = ut + ½at². This calculator makes this selection automatically.
Can SUVAT equations be used for free fall?
Yes. Free fall is the simplest SUVAT application. For a dropped object: u = 0, a = 9.81 m/s² (or −9.81 if you take upward as positive). For an object thrown upward: u is positive and a = −9.81 m/s². All five SUVAT equations apply, with g replacing a.
Do SUVAT equations work for deceleration?
Yes. Deceleration is simply negative acceleration. If a car brakes from 30 m/s to rest and you take the forward direction as positive, enter a as a negative number (e.g., −8 m/s²). The calculator handles negative acceleration correctly in all equations.
Can I use SUVAT for vertical and horizontal motion separately?
Yes. Projectile motion problems are solved by applying SUVAT equations independently to horizontal (a = 0, constant velocity) and vertical (a = g) components. Our Projectile Motion Calculator does exactly this — it applies SUVAT to both axes simultaneously.
Why does the quadratic equation sometimes give two answers?
Equations 2 and 3 are quadratic in time, meaning the mathematics produces two solutions. Both may be physically meaningful — for example, a ball thrown upward passes through a height of 5m twice: once on the way up and once on the way back down. The calculator displays both roots when both are positive and labels which corresponds to the ascending and descending phases.
What is the most common mistake with SUVAT equations?
The most common error is incorrect sign convention — particularly with gravity. If you take upward as positive, then g must be entered as −9.81 m/s², not +9.81. Entering the wrong sign for acceleration reverses the direction of motion in the calculation and produces nonsensical results. Always decide your positive direction before entering values.
Are SUVAT equations used in real engineering?
Yes, extensively. Automotive engineers use them for braking distance calculations and crash analysis. Aerospace engineers use them for takeoff and landing roll calculations. Civil engineers use them for elevator and escalator design. Roller coaster designers use them for launch track length calculations. Military engineers use them for ballistic trajectory baselines. Any system involving constant acceleration in a straight line is a SUVAT problem.
The Scientists Behind SUVAT
The Kinematics Behind This Calculator
Galileo Galilei (1564–1642)
Every SUVAT equation traces back to Galileo's inclined-plane experiments proving that falling bodies accelerate uniformly, and that distance from rest grows with the square of time.
Read his full biography →
Why Constant Acceleration Happens
Isaac Newton (1643–1727)
Newton's second law, F = ma, explains why a constant force produces the constant acceleration that every SUVAT equation assumes — the dynamics behind Galileo's kinematics.
Read his full biography →