SUVAT Equations: All 5 Formulas, Derivations & Examples

SUVAT Equations: The Complete Physics Guide (With Derivations & Solvers)

Every physics exam you will ever take in mechanics has one thing in common — it expects you to predict how objects move. How far, how fast, how long. The SUVAT equations are the tool that makes this possible. These five equations have been the backbone of motion problems since Galileo first rolled balls down ramps in the 1600s, and nearly 400 years later, they still appear on every GCSE, A-Level, and AP Physics paper.

What makes the SUVAT formulas so powerful is their simplicity. They connect just five everyday quantities — distance, speed at the start, speed at the end, acceleration, and time — into a tight set of relationships. Engineers at NASA used these same principles to calculate the trajectories of the Apollo missions. Crash-test labs use them daily to model vehicle collisions. And yet, each equation is short enough to fit on a sticky note.

This guide walks you through everything from the ground up — no prior knowledge needed. You will see where each equation comes from (not just what it looks like), learn a step-by-step method that works on any problem, and practise on real worked examples at three difficulty levels. We also cover the tricky parts that trip students up most: sign conventions, 2D motion, and knowing exactly when these equations stop working.

A Brief History of the SUVAT Equations

The story begins with Galileo Galilei in the early 1600s. He rolled balls down ramps in Padua, timed their descent, and discovered that falling distance is proportional to the square of time — the relationship we now write as s = ut + ½at². This overturned nearly two thousand years of Aristotle’s claim that heavier objects fall faster. Galileo lacked modern algebra, but the relationships he found are exactly the SUVAT equations we use today.

Isaac Newton built on Galileo’s work in his 1687 Principia Mathematica, where his second law (F = ma) explained why constant force produces constant acceleration. His development of calculus provided the formal tools to derive every SUVAT equation through differentiation and integration. The five-variable framework was standardised over the following centuries, and the acronym “SUVAT” itself became popular through British GCSE and A-Level syllabi — but the physics behind it has remained unchanged for over three hundred years.

Introduction to SUVAT Equations

What Are the SUVAT Equations?

Have you ever wondered how scientists predict exactly where a ball will land? Or how engineers know the braking distance of a car at 100 km/h? They use the SUVAT equations.

The word “SUVAT” is not a complicated science term. It is just an acronym. Each letter stands for one of the five variables used in these equations:

These equations are some of the most used equations and formulas in all of physics. They appear in almost every mechanics problem you will face.

The Fundamental Rule: When Can You Use SUVAT?

Here is the golden rule: you can only use the SUVAT equations when acceleration is constant (uniform). This means the acceleration does not change at any point during the motion.

When SUVAT works:

• A ball falling freely near Earth’s surface (acceleration due to gravity is constant at 9.81 m/s²).
• A car braking with steady deceleration on a flat road.
• A hockey puck sliding across smooth ice with constant friction.

When SUVAT does NOT work:

• A skydiver falling through air (air resistance increases with speed, so acceleration changes).
• A bungee jumper (the cord stretches more the further they fall, changing the acceleration).
• A car in stop-and-go traffic (acceleration keeps changing).

Most SUVAT problems deal with 1D linear motion — movement along a single straight line. But you can also use these equations in 2D by breaking the motion into separate x- and y-components. We cover that later in the projectile motion section.

The 5 SUVAT Equations — Master Table

Here are all five SUVAT formulas in one place. Each equation leaves out one variable. This is what makes them so useful — you pick the equation that does not need the variable you do not know.

Graphical Foundations & Sign Conventions

Motion Graphs (Visual Intuition)

Graphs are one of the best ways to understand motion. There are three main types of motion graphs you should know.

Displacement–Time (s–t) Graphs

This graph plots displacement on the y-axis and time on the x-axis.

• The slope (steepness) of the line at any point gives you the velocity.
• A straight line means constant velocity.
• A curved line means the velocity is changing (the object is accelerating).
• A flat horizontal line means the object is not moving.

Velocity–Time (v–t) Graphs

This graph plots velocity on the y-axis and time on the x-axis.

• The slope of the line gives you the acceleration.
• The area under the line gives you the displacement.
• A straight horizontal line means constant velocity (zero acceleration).
• A straight line going upward means constant positive acceleration.
• A straight line going downward means deceleration (or acceleration in the negative direction).

Acceleration–Time (a–t) Graphs

This graph plots acceleration on the y-axis and time on the x-axis.

• For constant acceleration, this is just a flat horizontal line.
• The area under the line gives you the change in velocity (Δv).

Vectors vs. Scalars: Mastering Sign Conventions

One of the trickiest parts of SUVAT problems is getting the signs right. Here is how to handle them.

First, you must pick a positive direction. The most common choices are:

• Horizontal motion: Right = Positive (+), Left = Negative (−)
• Vertical motion: Up = Positive (+), Down = Negative (−)

Once you pick a direction, stick with it for the entire problem.

Handling free-fall acceleration:

Earth’s gravitational acceleration is about 9.81 m/s². If you choose “up” as positive, then the acceleration due to gravity is:

The negative sign means gravity pulls downward, which is the negative direction.

Deceleration vs. negative acceleration:

These are not always the same thing!

• Deceleration means the object is slowing down. If a car moves to the right (+) and brakes, then a = −3 m/s² (negative, slowing down).
• Speeding up in the negative direction means the object is getting faster while moving in the negative direction. If a ball falls downward (−) and gravity makes it go faster, then a = −9.81 m/s². The acceleration is negative, but the ball is speeding up, not slowing down.

Understanding forces and how they relate to these signs is crucial for solving physics problems correctly.

How to Derive Every SUVAT Equation

Understanding where these equations come from makes them much easier to remember and use. There are three ways to derive them.

Method 1: Algebraic Derivations (Standard Physics Level)

Deriving Equation 1: v = u + at

Start with the definition of acceleration:

Multiply both sides by t:

at = v − u

Add u to both sides:

This equation simply says: “Your final velocity equals your starting velocity plus whatever velocity you gained from accelerating.”

Deriving Equation 2: s = ½(u + v)t

When acceleration is constant, the average velocity is simply the midpoint of u and v:

Displacement equals average velocity times time:

Deriving Equation 3: s = ut + ½at²

Take Equation 1: v = u + at. Substitute this into Equation 2:

s = ½(u + [u + at])t

s = ½(2u + at)t

This is one of the most commonly used SUVAT formulas. It tells you the displacement when you know the starting velocity, acceleration, and time.

Deriving Equation 4: v² = u² + 2as

From Equation 1: v = u + at, so t = (v − u) / a.

Substitute this into Equation 2:

s = ½(u + v) × (v − u) / a

s = (u + v)(v − u) / 2a

s = (v² − u²) / 2a

Multiply both sides by 2a:

2as = v² − u²

Rearrange:

This equation is especially useful when time is not given in the problem.

The Energy Connection: Why v² = u² + 2as Is Really About Energy

Deriving Equation 5: s = vt − ½at²

From Equation 1: u = v − at. Substitute into Equation 3:

s = (v − at)t + ½at²

s = vt − at² + ½at²

This equation is useful when you do not know the initial velocity.

Method 2: Graphical Derivation (Using Velocity–Time Graphs)

You can derive the SUVAT equations by looking at the area under a v–t graph.

Draw a v–t graph for an object with constant acceleration. It starts at velocity u and ends at velocity v after time t. This forms a trapezoid (a shape with one pair of parallel sides).

The area of this trapezoid gives the displacement:

• Rectangle area = u × t = ut (the displacement from the initial velocity alone)
• Triangle area = ½ × base × height = ½ × t × (v − u) = ½ × t × at = ½at²
• Total area = ut + ½at² = s

This is Equation 2. The same graphical approach confirms all five SUVAT equations.

Method 3: Calculus Derivations (University Level)

If you know calculus, the SUVAT equations follow naturally from integration.

Starting from Differentiation: The Fundamental Definitions

Before we integrate anything, we need to understand where the quantities come from. Calculus defines velocity and acceleration as derivatives of the quantities before them.

Velocity is the rate of change of displacement with respect to time:

This means if you know the displacement of an object as a function of time, s(t), you can find its velocity at any instant by differentiating.

Acceleration is the rate of change of velocity with respect to time:

These two definitions form the mathematical backbone of all kinematics. Differentiation moves you down the chain — from displacement to velocity to acceleration. Integration moves you back up — from acceleration to velocity to displacement.

Step 1: Integrate acceleration to get velocity.

Since acceleration is constant:

v(t) = ∫a dt = at + C

At t = 0, v = u, so C = u:

This is Equation 1.

Step 2: Integrate velocity to get displacement.

s(t) = ∫(u + at) dt = ut + ½at² + C

At t = 0, s = 0, so C = 0:

This is Equation 2.

Calculus gives us the same results as algebra and graphs. It also shows why SUVAT only works for constant acceleration — when acceleration changes with time, the integrals produce different, more complex equations.

The 5-Step SUVAT Problem-Solving Framework

Follow these five steps every time you solve a SUVAT problem. This method works for almost any constant acceleration question.

Step 1: Draw a Diagram & Choose a Positive Direction

Sketch the situation: Mark the starting point, the direction of motion, and the direction of acceleration. Then pick a positive direction and label it clearly.

Tip: For vertical problems, “up = positive” is the standard choice.

Step 2: List the SUVAT Variables (s, u, v, a, t)

Write down all five SUVAT variables in a column. Fill in the values you know.

Step 3: Identify the 3 Knowns and the 1 Target Variable

You need three known values to solve a problem. Identify which variable you need to find (the target). That gives you four of the five variables.

Step 4: Pick the Equation That Excludes the 5th (Missing) Variable

Look at the master table. Find the equation that does not contain the variable you neither know nor need. This is the equation that will work.

Step 5: Rearrange, Substitute Values with Signs, and Check Units

Plug your known values into the equation. Be careful with signs. Solve for the target variable. Then check that your answer has the correct basic units.

Worked Examples by Difficulty Level

Level 1: Standard Horizontal Motion

Example 1: Braking Automobile — Stopping Distance

A car is travelling at 25 m/s on a straight road. The driver brakes, causing a uniform deceleration of 5 m/s². How far does the car travel before stopping?

Step 1: Direction of motion is positive (to the right). Deceleration acts opposite, so a is negative.

Step 2: List variables:

• s = ? (this is what we want)
• u = 25 m/s
• v = 0 m/s (the car stops)
• a = −5 m/s²
• t = not given

Step 3: We know u, v, and a. We want s. We do not know t.

Step 4: The equation that excludes t is Equation 5: v² = u² + 2as.

Step 5: Substitute:

0² = 25² + 2(−5)(s)

0 = 625 − 10s

10s = 625

The car travels 62.5 metres before stopping. This is roughly the length of six buses parked end to end.

Level 2: Vertical Motion Under Gravity

Example 2: Vertical Launch — Ball Thrown Upward

A ball is thrown straight up with an initial velocity of 20 m/s. Taking g = 9.81 m/s², find: (a) the maximum height, (b) the time to reach the peak, and (c) the total flight time.

Setup: Take up as positive. So u = +20 m/s and a = −9.81 m/s².

(a) Maximum height:

At the peak, v = 0 m/s. Time is unknown. Use Equation 5:

v² = u² + 2as

0 = 20² + 2(−9.81)(s)

0 = 400 − 19.62s

s = 400 ÷ 19.62

(b) Time to reach the peak:

Use Equation 1: v = u + at

0 = 20 + (−9.81)t

9.81t = 20

(c) Total flight time:

By symmetry, the ball takes the same time to go up and come back down (if it returns to the same height). So:

Total time = 2 × 2.04 = 4.08 s

Understanding how gravity acts on objects is essential for vertical motion problems like this.

Example 3: Cliff Drop — Impact Velocity

A stone is dropped from rest off a 45 m high cliff. What is its velocity just before hitting the ground? (Take g = 9.81 m/s²)

Setup: Take down as positive (since that is the direction of motion). So u = 0, a = +9.81 m/s², s = 45 m.

Use Equation 5: v² = u² + 2as

v² = 0 + 2(9.81)(45)

v² = 882.9

Level 3: Multi-Stage & Relative Motion Problems

Example 4: Overtaking / Catch-up Scenario

Car A passes a point at a constant velocity of 15 m/s. At the same instant, Car B starts from rest at the same point with a constant acceleration of 3 m/s². How long does it take Car B to catch up with Car A?

Both cars travel the same displacement s in the same time t.

Car A (constant velocity, so a = 0):

s = ut + ½at² = 15t + 0 = 15t

Car B (from rest, u = 0):

s = 0 + ½(3)t² = 1.5t²

Set equal: 15t = 1.5t²

Divide both sides by t (t ≠ 0): 15 = 1.5t

At that moment, both cars have covered s = 15 × 10 = 150 m.

Example 5: Two-Stage Rocket

A model rocket accelerates vertically at 15 m/s² for 8 seconds, then the engine cuts out. Find (a) the velocity when the engine stops, (b) the extra height gained after engine cut-out, and (c) the total maximum height.

Stage 1: Engine on (constant acceleration upward)

• u₁ = 0 (starts from rest)
• a₁ = 15 m/s²
• t₁ = 8 s

Velocity at engine cut-out:

v₁ = u₁ + a₁t₁ = 0 + 15(8) = 120 m/s

Height during Stage 1:

s₁ = u₁t₁ + ½a₁t₁² = 0 + ½(15)(64) = 480 m

Stage 2: Engine off (free-fall under gravity only)

Now u₂ = 120 m/s (velocity carried over), a₂ = −9.81 m/s² (gravity slows it down), v₂ = 0 at maximum height.

Extra height after engine cut-out:

v₂² = u₂² + 2a₂s₂

0 = 120² + 2(−9.81)(s₂)

0 = 14400 − 19.62s₂

s₂ = 14400 ÷ 19.62 = 734 m

That is over 1.2 km high — taller than the Burj Khalifa (828 m), the world’s tallest building!

This is a classic multi-stage problem. You need to apply Newton’s laws of motion to understand why the rocket keeps rising after the engine stops.

SUVAT in 2D Motion & Advanced Applications

Projectile Motion (Decomposing Vectors)

When an object is launched at an angle (like a football kick or a cannonball), it moves in two directions at once: horizontal and vertical. This is called projectile motion.

The trick is to split the motion into two independent parts using the SUVAT equations:

Horizontal (x-axis):

• There is no horizontal acceleration (ignoring air resistance): aₓ = 0
• So horizontal velocity stays constant: vₓ = uₓ
• Horizontal displacement: sₓ = uₓ × t

Vertical (y-axis):

• The only acceleration is gravity: aᵧ = −g = −9.81 m/s²
• Use any SUVAT equation with this acceleration.

The link between the two: Time (t) is the same for both directions. You often solve for t in one direction and use it in the other.

You can calculate projectile motion problems using our projectile motion calculator.

Motion on Inclined Planes

When an object slides down a frictionless slope at angle θ, gravity is not fully accelerating it. Only the component of gravity along the slope matters:

Vector SUVAT (Matrix / Bold Notation)

At the university level, SUVAT equations are written using vectors. This lets you handle 2D and 3D motion in one compact equation:

Here, r is the position vector, r₀ is the starting position, u is the initial velocity vector, and a is the acceleration vector. Each of these has both x and y components (and z for 3D).

This is the same as writing two separate SUVAT equations — one for each axis — but combined into one neat form. This notation is commonly used in engineering physics.

Worked Example: SUVAT with i, j Unit Vectors

A-Level and university problems often express SUVAT variables as vectors using unit vector notation. Here is how to handle them.

A particle has initial velocity u = (3i + 4j) m/s and constant acceleration a = (2i − 1j) m/s². Find its velocity, speed, and position at t = 3 s.

Velocity — use v = u + at, applied to each component separately:

v = (3i + 4j) + (2i − 1j)(3)

v = (3 + 6)i + (4 − 3)j = 9i + 1j m/s

Speed — find the magnitude of the velocity vector:

|v| = √(9² + 1²) = √82 = 9.06 m/s

Position — use r = r₀ + ut + ½at², assuming the particle starts at the origin (r₀ = 0):

r = (3i + 4j)(3) + ½(2i − 1j)(9)

r = (9 + 9)i + (12 − 4.5)j = 18i + 7.5j m

The process is the same as regular SUVAT — you just apply each equation to the i and j components independently and recombine at the end.

When SUVAT Fails (Limitations & Boundary Cases)

The SUVAT equations are powerful, but they have limits. Here are the main situations where they break down.

Non-Constant Acceleration (Jerk & Drag Force)

Air resistance: When objects move through air, the drag force depends on velocity (often F_drag ∝ v²). This means acceleration changes every instant. A skydiver, for example, accelerates at 9.81 m/s² right after jumping, but this drops to zero when they reach terminal velocity (about 53 m/s or 190 km/h in a belly-down position). SUVAT cannot handle this changing acceleration.

Simple harmonic motion: Springs and pendulums produce an acceleration that depends on position (F ∝ −x). The further you stretch a spring, the stronger it pulls back. The acceleration is never constant, so SUVAT does not apply.

Variable mass systems: A rocket burning fuel gets lighter over time. As its mass decreases, the same engine thrust produces a larger acceleration. The acceleration changes continuously, making SUVAT inappropriate.

Circular & Non-Linear Paths

Objects moving in circles have centripetal acceleration pointing toward the centre. This acceleration constantly changes direction, even if its magnitude stays the same. Since SUVAT assumes constant acceleration in both magnitude and direction along a straight line, it does not apply to circular motion.

However, if an object moves along a circular path and speeds up or slows down, it also has a tangential acceleration along its path. You can use SUVAT for just this tangential component, as long as it stays constant.

Relativistic Kinematics

At extremely high speeds — close to the speed of light (about 300,000,000 m/s or 3 × 10⁸ m/s) — Newton’s laws and the SUVAT equations stop being accurate. At these speeds, effects from Einstein’s Special Relativity take over. Time slows down, lengths contract, and mass effectively increases. Particles in accelerators like CERN’s Large Hadron Collider routinely reach 99.9999% of the speed of light. For them, SUVAT is completely inadequate, and relativistic equations must be used instead.

For the everyday speeds we experience (cars, balls, planes), SUVAT works perfectly.

You can learn more about the deeper physics behind these limits in our articles on quantum physics and nuclear physics.

Common SUVAT Mistakes to Avoid — Checklist

Here are the five most common errors students make with SUVAT equations. Avoid these, and you will solve problems correctly much more often.

For a more complete list of key physics terms and definitions, visit our glossary page.

Interactive Practice Tools

Ready to test your understanding? Try solving SUVAT problems using our online calculators:

• Projectile Motion Calculator — Enter launch angle and speed to see the full trajectory.
• Momentum Calculator — See how velocity and mass connect to momentum through Newton’s second law.

Understanding SUVAT equations also helps with many other physics topics, including electricity, waves, light, and power. The problem-solving framework you learn here applies across all of physics.

Conclusion

The SUVAT equations are five of the most important tools in physics. They let you solve any constant-acceleration problem using just three known values. Whether a car is braking on a highway, a ball is flying through the air, or a rocket is launching into the sky, these five equations can tell you exactly what happens.

Here is what we covered:

The SUVAT meaning is simple — it stands for Displacement (S), Initial Velocity (U), Final Velocity (V), Acceleration (A), and Time (T). These five variables describe any straight-line motion with constant acceleration.

The five SUVAT formulas are v = u + at, s = ut + ½at², s = vt − ½at², s = ½(u + v)t, and v² = u² + 2as. Each one leaves out a different variable, so you can always find the right equation for any problem.

We showed three ways to derive every SUVAT equation — using algebra, graphs, and calculus. We walked through a 5-step problem-solving framework that works every time. And we solved five worked examples, from a simple braking car to a two-stage rocket reaching over 1,200 metres.

We also covered how to extend SUVAT into 2D for projectile motion and inclined planes, and we looked at the cases where SUVAT fails — air resistance, circular motion, and speeds near the speed of light.

Frequently Asked Questions

The Scientists Behind the SUVAT Equations

Who Formalised the Mathematics

Isaac Newton (1643–1727)

Newton’s laws of motion and his invention of calculus provided the formal tools to derive every SUVAT equation. His 1687 Principia Mathematica remains the foundation of classical mechanics.

Read his full biography →

Who Discovered the Relationships

Galileo Galilei (1564–1642)

Galileo’s inclined plane experiments first revealed that falling distance is proportional to the square of time — the physical relationship behind s = ut + ½at², decades before Newton gave it mathematical form.

Read his full biography →