SUVAT Equations Explained: Powerful Motion Formulas 2026

Motion problems can look complicated when several quantities are changing at the same time. An object may have an initial velocity, accelerate for several seconds, travel a certain distance, and reach a different final velocity. The SUVAT equations provide a simple way to connect these quantities when acceleration is constant.

The name SUVAT comes from the five variables used in the equations: s for displacement, u for initial velocity, v for final velocity, a for acceleration, and t for time. These equations are also called the kinematic equations and are widely used in physics to solve problems involving uniformly accelerated motion.

Whether you are studying a car accelerating on a straight road, an object in free fall, or a projectile’s vertical motion, knowing which SUVAT equation to use can make the calculation much easier.

What Are the SUVAT Equations?

The SUVAT equations are five equations used to describe motion with constant acceleration.

The five variables are:

• s = displacement
• u = initial velocity
• v = final velocity
• a = acceleration
• t = time

The five standard equations are:

v = u + at

s = ut + ½at²

s = vt − ½at²

s = ½(u + v)t

v² = u² + 2as

These equations allow you to calculate an unknown quantity when enough other information is known.

What Does SUVAT Stand For?

The word SUVAT is simply an acronym.

Letter	Quantity	SI Unit
S	Displacement	m
U	Initial velocity	m/s
V	Final velocity	m/s
A	Acceleration	m/s²
T	Time	s

The important point is that s represents displacement rather than simply distance.

Displacement includes direction, so signs matter when solving SUVAT problems.

The First SUVAT Equation: v = u + at

The first equation is:

v = u + at

It connects:

• Initial velocity
• Final velocity
• Acceleration
• Time

This equation is particularly useful when displacement is not needed.

Example

A car starts at 5 m/s and accelerates at 3 m/s² for 4 seconds. Find its final velocity.

Given:

u = 5 m/s

a = 3 m/s²

t = 4 s

Use:

v = u + at

Substitute:

v = 5 + (3)(4)

v = 17 m/s

Therefore, the final velocity is:

17 m/s

The Second SUVAT Equation: s = ut + ½at²

The second equation is:

s = ut + ½at²

It connects:

• Displacement
• Initial velocity
• Acceleration
• Time

It is useful when you know initial velocity, acceleration, and time and want to find displacement.

Example

A car travels at an initial velocity of 10 m/s and accelerates at 2 m/s² for 5 seconds. Find its displacement.

Given:

u = 10 m/s

a = 2 m/s²

t = 5 s

Use:

s = ut + ½at²

Substitute:

s = (10)(5) + ½(2)(5²)

s = 50 + 25

s = 75 m

The displacement is:

75 m

The Third SUVAT Equation: s = vt − ½at²

The third equation is:

s = vt − ½at²

It uses:

• Displacement
• Final velocity
• Acceleration
• Time

This version is useful when the final velocity is known instead of the initial velocity.

Example

An object has a final velocity of 20 m/s after accelerating at 4 m/s² for 3 seconds. Find its displacement.

Use:

s = vt − ½at²

Substitute:

s = (20)(3) − ½(4)(3²)

s = 60 − 18

s = 42 m

Therefore:

s = 42 m

The Fourth SUVAT Equation: s = ½(u + v)t

The fourth equation is:

s = ½(u + v)t

This equation uses the average velocity:

Average velocity = (u + v)/2

Therefore:

s = average velocity × time

This equation is useful when you know initial velocity, final velocity, and time.

Example

A vehicle increases its velocity from 8 m/s to 20 m/s in 6 seconds. Find its displacement.

Use:

s = ½(u + v)t

Substitute:

s = ½(8 + 20)(6)

s = ½(28)(6)

s = 84 m

The displacement is:

84 m

The Fifth SUVAT Equation: v² = u² + 2as

The fifth equation is:

v² = u² + 2as

This equation is especially useful when time is not given.

It connects:

• Final velocity
• Initial velocity
• Acceleration
• Displacement

Example

A car starts at 10 m/s and accelerates at 3 m/s² over a distance of 50 m. Find its final velocity.

Given:

u = 10 m/s

a = 3 m/s²

s = 50 m

Use:

v² = u² + 2as

Substitute:

v² = 10² + 2(3)(50)

v² = 100 + 300

v² = 400

Therefore:

v = 20 m/s

The final velocity is:

20 m/s

How to Choose the Correct SUVAT Equation

Choosing the correct equation is often the hardest part of a SUVAT problem.

A useful method is to list all the known quantities first.

For example:

u = ?

v = 20 m/s

a = 3 m/s²

s = 50 m

t = ?

You then choose an equation containing the known values and the unknown you need.

For example:

v² = u² + 2as

contains v, u, a, s.

If all except u are known, it may be appropriate.

A Simple SUVAT Equation Table

Equation	Best Used When
v = u + at	Displacement is not needed
s = ut + ½at²	Final velocity is not known
s = vt − ½at²	Initial velocity is not known
s = ½(u + v)t	Initial and final velocities are known
v² = u² + 2as	Time is not known

The best equation is usually the one containing the quantities you already know and the quantity you need to calculate.

Worked Example: Full SUVAT Problem

A train is traveling at 12 m/s and accelerates uniformly at 2 m/s² for 8 seconds.

Find:

1. Its final velocity
2. Its displacement

Step 1: Identify the variables

u = 12 m/s

a = 2 m/s²

t = 8 s

v = ?

s = ?

Step 2: Find final velocity

Use:

v = u + at

v = 12 + (2)(8)

v = 28 m/s

Step 3: Find displacement

Use:

s = ut + ½at²

s = (12)(8) + ½(2)(8²)

s = 96 + 64

s = 160 m

Therefore:

Final velocity = 28 m/s

Displacement = 160 m

SUVAT Equations and Free Fall

The SUVAT equations are particularly useful for objects falling under gravity.

Near Earth’s surface, ignoring air resistance:

a = g ≈ 9.8 m/s²

Depending on the direction you choose as positive, acceleration due to gravity can be:

+9.8 m/s²

or:

−9.8 m/s²

The sign depends entirely on your chosen coordinate system.

Worked Example: Falling Object

An object is dropped from rest and falls for 3 seconds. Ignore air resistance. Find its final velocity.

Since the object is dropped:

u = 0

Take downward as positive:

a = 9.8 m/s²

t = 3 s

Use:

v = u + at

v = 0 + (9.8)(3)

v = 29.4 m/s

The final velocity is:

29.4 m/s downward

Finding Distance Fallen

Using the same example:

u = 0

a = 9.8 m/s²

t = 3 s

Use:

s = ut + ½at²

s = (0)(3) + ½(9.8)(3²)

s = 4.9 × 9

s = 44.1 m

The object falls approximately:

44.1 m

SUVAT and Projectile Motion

SUVAT equations can also be used for the vertical component of projectile motion.

For a projectile:

uᵧ = u sinθ

and:

aᵧ = −g

The vertical motion can then be analyzed using the SUVAT equations.

For example:

vᵧ = uᵧ − gt

and:

y = uᵧt − ½gt²

The horizontal component has constant velocity when air resistance is ignored:

x = uₓt

This separation of horizontal and vertical motion is one of the most useful ideas in projectile problems.

Why Is Acceleration Constant?

The SUVAT equations only apply when acceleration is constant.

For example, near Earth’s surface, gravitational acceleration is approximately constant:

g = 9.8 m/s²

This makes the equations suitable for many basic free-fall problems.

However, if acceleration changes significantly with time, the standard SUVAT equations cannot be applied directly over the entire motion.

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Displacement vs Distance

One common source of mistakes is confusing displacement with distance.

Distance is the total path length traveled.

Displacement is the change in position from the starting point to the ending point.

For example, if you walk 10 m forward and then 10 m backward:

Distance = 20 m

but:

Displacement = 0 m

SUVAT equations use displacement, represented by s.

Velocity vs Speed

SUVAT problems also use velocity rather than simply speed.

Velocity includes direction.

For example:

+10 m/s

and:

−10 m/s

have the same speed but opposite velocities.

The sign becomes especially important in vertical motion.

Acceleration and Signs

Acceleration is also directional.

If upward is positive, gravity is:

a = −9.8 m/s²

If downward is positive, gravity is:

a = +9.8 m/s²

Either convention works as long as you use it consistently.

A useful strategy is to choose a positive direction at the beginning of the problem and keep that choice throughout.

Deriving the First SUVAT Equation

The first equation comes directly from the definition of constant acceleration.

Acceleration is:

a = Δv/Δt

Therefore:

a = (v − u)/t

Multiply by t:

at = v − u

Rearrange:

v = u + at

So the first SUVAT equation follows directly from the definition of acceleration.

Deriving the Fourth SUVAT Equation

For constant acceleration, average velocity is:

average velocity = (u + v)/2

Displacement equals average velocity multiplied by time:

s = average velocity × t

Therefore:

s = [(u + v)/2]t

So:

s = ½(u + v)t

This gives the fourth SUVAT equation.

Deriving v² = u² + 2as

Start with:

v = u + at

Rearrange:

t = (v − u)/a

Using:

s = ½(u + v)t

substitute the expression for time:

s = ½(u + v)(v − u)/a

Multiply both sides by 2a:

2as = (u + v)(v − u)

Using the difference of squares:

(u + v)(v − u) = v² − u²

Therefore:

2as = v² − u²

Rearranging gives:

v² = u² + 2as

This explains why the fifth SUVAT equation does not contain time.

When Should You Not Use SUVAT Equations?

The standard SUVAT equations should not be used directly when acceleration is not constant throughout the motion.

For example, if an object’s acceleration changes continuously and significantly, more advanced methods such as calculus or numerical analysis may be required.

The equations also depend on the motion being described appropriately in one dimension or along a single component with constant acceleration.

Common SUVAT Mistakes

1. Using Distance Instead of Displacement

Remember:

s = displacement

not necessarily total distance traveled.

2. Forgetting Units

Use:

• Displacement: m
• Velocity: m/s
• Acceleration: m/s²
• Time: s

3. Using the Wrong Sign

Choose a positive direction and remain consistent.

4. Using SUVAT With Variable Acceleration

The standard equations require constant acceleration.

5. Forgetting to Square Velocity

In:

v² = u² + 2as

both v and u are squared.

6. Taking the Wrong Square Root

When solving for velocity, mathematical solutions can produce positive or negative values. The physically correct sign depends on the direction of motion.

SUVAT Formula Sheet

Here are all five standard equations together:

1. v = u + at

2. s = ut + ½at²

3. s = vt − ½at²

4. s = ½(u + v)t

5. v² = u² + 2as

Remember:

s = displacement

u = initial velocity

v = final velocity

a = acceleration

t = time

Frequently Asked Questions

Conclusion

The SUVAT equations are some of the most useful formulas for analyzing motion with constant acceleration. They connect displacement, initial velocity, final velocity, acceleration, and time in five closely related equations.

The five formulas are:

v = u + at

s = ut + ½at²

s = vt − ½at²

s = ½(u + v)t

v² = u² + 2as

The key to using them successfully is not simply memorizing every equation. First identify the known quantities, determine the unknown quantity, choose an equation containing the relevant variables, and make sure acceleration is constant.

With practice, the SUVAT equations can be used to solve a wide range of problems involving cars, falling objects, projectiles, and other forms of uniformly accelerated motion.

#Suvat Equations