Last Updated: September 5, 2026

Quick Summary & Key Takeaways

Target Audience: High school and university physics students, including IB Physics, IGCSE, A-Levels, AP Physics, and students studying introductory mechanics and kinematics.

What are SUVAT equations?

SUVAT equations are five kinematic equations of motion used to solve one-dimensional motion when acceleration is constant. The word SUVAT comes from the five variables used in the equations:

• S = displacement
• U = initial velocity
• V = final velocity
• A = acceleration
• T = time

The 5 SUVAT equations

1. (v = u + at) — does not contain (s)
2. (s = ut + \frac{1}{2}at^2) — does not contain (v)
3. (v^2 = u^2 + 2as) — does not contain (t)
4. (s = \frac{1}{2}(u+v)t) — does not contain (a)
5. (s = vt – \frac{1}{2}at^2) — does not contain (u)

SUVAT equations are especially useful for constant-acceleration motion, including braking, free fall, vertical projection, straight-line acceleration, and the horizontal or vertical components of projectile motion.

Important: SUVAT equations apply when acceleration is constant. If acceleration varies with time or position, calculus-based kinematics is required.

Introduction to SUVAT Kinematic Equations

Understanding SUVAT equations is essential for solving many classical mechanics and kinematics problems. These five formulas describe the relationship between displacement, initial velocity, final velocity, acceleration, and time when an object moves with constant acceleration.

Whether you are solving an IGCSE, A-Level, IB Physics, or AP Physics problem, SUVAT provides a systematic way to determine an unknown quantity from the information given.

For example, SUVAT can be used to calculate how far a car travels while braking, how fast an object falls under gravity, the maximum height of a vertically thrown ball, or the final velocity of an accelerating train.

SUVAT equations are also closely connected to Newton’s laws of motion. Newton’s Second Law, (F=ma), can be used to determine acceleration from net force, while SUVAT can then describe the resulting motion.

For students studying broader mechanics, these equations also connect with projectile motion, inclined-plane motion, velocity-time graphs, force and acceleration, and energy concepts.

This SUVAT guide is designed for students studying GCSE, IGCSE, A-Level, IB, AP Physics, and introductory mechanics, as well as learners reviewing basic kinematics for engineering and STEM courses. Use these equations when a motion problem involves known values such as displacement, initial velocity, final velocity, acceleration, or time and the acceleration is constant. Typical applications include braking and stopping distances, cars and trains, free fall, vertical throws, projectile-motion components, and motion on frictionless slopes.

What Does SUVAT Mean?

The term SUVAT is an abbreviation based on five variables used in constant-acceleration kinematics.

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

So, when someone asks “what does SUVAT stand for?”, the answer is:

S = displacement, U = initial velocity, V = final velocity, A = acceleration, and T = time.

The term SUVAT is therefore not a physical law by itself. It is a convenient name for a group of five kinematic equations of motion.

What Are the 5 SUVAT Equations?

The five standard SUVAT formulas are:

1. SUVAT Equation Without (s)

[
v = u + at
]

This equation is useful when displacement (s) is not needed or is unknown.

It can also be rearranged to find time or acceleration:

[
t = \frac{v-u}{a}
]

[
a = \frac{v-u}{t}
]

2. SUVAT Equation Without (v)

[
s = ut + \frac{1}{2}at^2
]

Use this equation when final velocity (v) is not required.

3. SUVAT Equation Without (t)

[
v^2 = u^2 + 2as
]

This is particularly useful when time is not given.

It can also be rearranged as:

[
s = \frac{v^2-u^2}{2a}
]

or

[
a = \frac{v^2-u^2}{2s}
]

4. SUVAT Equation Without (a)

[
s = \frac{1}{2}(u+v)t
]

This equation uses average velocity and is useful when acceleration does not need to be included directly.

5. SUVAT Equation Without (u)

[
s = vt-\frac{1}{2}at^2
]

This equation is useful when the initial velocity (u) is the variable that is not required.

All SUVAT Equations at a Glance

If you are looking for an all SUVAT equations list, use this table:

Equation	Missing Variable
(v=u+at)	(s)
(s=ut+\frac12at^2)	(v)
(v^2=u^2+2as)	(t)
(s=\frac12(u+v)t)	(a)
(s=vt-\frac12at^2)	(u)

A useful exam strategy is to identify the variable that is not available and not required, then choose the SUVAT equation that excludes it.

SUVAT Equations Rearranged

SUVAT formulas are often rearranged depending on what the question asks.

Finding acceleration

From:

[
v=u+at
]

[
a=\frac{v-u}{t}
]

From:

[
v^2=u^2+2as
]

[
a=\frac{v^2-u^2}{2s}
]

Finding time

From:

[
v=u+at
]

[
t=\frac{v-u}{a}
]

From:

[
s=\frac12(u+v)t
]

[
t=\frac{2s}{u+v}
]

Finding displacement

[
s=ut+\frac12at^2
]

or

[
s=\frac12(u+v)t
]

or

[
s=vt-\frac12at^2
]

Finding final velocity

[
v=u+at
]

or

[
v^2=u^2+2as
]

Therefore, when students search for SUVAT equations rearranged, the important point is that the five equations can be algebraically rearranged to solve for the unknown variable.

How Many SUVAT Equations Are There?

There are five standard SUVAT equations for one-dimensional motion with constant acceleration.

They are not five unrelated laws. They are different mathematical relationships derived from the definitions of velocity, acceleration, and displacement under uniform acceleration.

You may see slightly different rearranged versions in textbooks or exam papers, but they are mathematically equivalent to the five standard equations.

How to Remember the SUVAT Equations

A practical way to remember SUVAT is to first memorize the five-variable structure:

S — U — V — A — T

Then remember the five equations according to the variable they leave out:

• No (S): (v=u+at)
• No (V): (s=ut+\frac12at^2)
• No (T): (v^2=u^2+2as)
• No (A): (s=\frac12(u+v)t)
• No (U): (s=vt-\frac12at^2)

This method is generally more useful than trying to memorize the formulas as isolated expressions.

SUVAT Equations and Their Physical Meaning

Variable	Physical meaning
(s)	Displacement from the starting position
(u)	Velocity at the beginning of the time interval
(v)	Velocity at the end of the time interval
(a)	Constant acceleration
(t)	Elapsed time

Displacement vs Distance

The (s) in SUVAT represents displacement, not necessarily total distance travelled.

Displacement is a vector quantity describing the change in position. Distance is the total path length.

For example, if an object travels 10 m forward and then 10 m back to its starting position:

• Distance = 20 m
• Displacement = 0 m

This distinction is important when solving SUVAT problems involving a change in direction.

SUVAT Acceleration Formula

The most direct SUVAT acceleration formula is:

[
a=\frac{v-u}{t}
]

where:

• (a) = acceleration in m/s²
• (v) = final velocity in m/s
• (u) = initial velocity in m/s
• (t) = time in seconds

Acceleration can be positive or negative depending on the chosen direction.

A negative acceleration does not automatically mean that an object is moving backward. It means that the acceleration vector points in the negative direction of the chosen coordinate system.

SUVAT Equations of Motion

SUVAT equations are commonly called equations of motion because they describe the motion of an object under constant acceleration.

They are especially useful in:

• straight-line kinematics
• vehicle acceleration and braking
• free-fall problems
• vertical motion
• projectile motion components
• inclined planes
• introductory mechanics
• IB, IGCSE, A-Level, and AP Physics problems

The essential condition is always the same:

[
a=\text{constant}
]

Deriving the SUVAT Equations

Understanding SUVAT derivations can make the formulas easier to remember.

The derivations begin with two basic definitions:

[
v=\frac{ds}{dt}
]

and

[
a=\frac{dv}{dt}
]

For constant acceleration, the five standard equations follow from these relationships.

Derivation 1: (v=u+at)

Acceleration is:

[
a=\frac{v-u}{t}
]

Multiply by (t):

[
at=v-u
]

Therefore:

[
\boxed{v=u+at}
]

Derivation 2: (s=\frac12(u+v)t)

For constant acceleration, average velocity is:

[
\bar v=\frac{u+v}{2}
]

Since displacement equals average velocity multiplied by time:

[
s=\bar vt
]

Therefore:

[
\boxed{s=\frac12(u+v)t}
]

Derivation 3: (s=ut+\frac12at^2)

Starting with:

[
s=\frac12(u+v)t
]

Substitute:

[
v=u+at
]

Then:

[
s=\frac12[u+(u+at)]t
]

[
s=\frac12(2u+at)t
]

Therefore:

[
\boxed{s=ut+\frac12at^2}
]

Derivation 4: (v^2=u^2+2as)

From:

[
v=u+at
]

we obtain:

[
t=\frac{v-u}{a}
]

Substitute this into:

[
s=\frac12(u+v)t
]

[
s=\frac12(u+v)\frac{v-u}{a}
]

Using difference of squares:

[
s=\frac{v^2-u^2}{2a}
]

Therefore:

[
\boxed{v^2=u^2+2as}
]

Derivation 5: (s=vt-\frac12at^2)

From:

[
v=u+at
]

we get:

[
u=v-at
]

Substitute into:

[
s=ut+\frac12at^2
]

[
s=(v-at)t+\frac12at^2
]

Therefore:

[
\boxed{s=vt-\frac12at^2}
]

Deriving SUVAT Equations Using Calculus

The SUVAT equations can also be derived directly using calculus.

Because:

[
a=\frac{dv}{dt}
]

for constant acceleration:

[
dv=a,dt
]

Integrating from initial velocity (u) to final velocity (v):

[
\int_u^v dv=\int_0^t a,dt
]

gives:

[
v-u=at
]

and therefore:

[
v=u+at
]

Similarly, because:

[
v=\frac{ds}{dt}
]

we have:

[
ds=v,dt
]

Substituting:

[
v=u+at
]

and integrating:

[
s=\int_0^t(u+at),dt
]

gives:

[
s=ut+\frac12at^2
]

The other SUVAT equations can then be obtained algebraically.

This is why SUVAT equations and calculus are closely connected: the equations are specific solutions for the special case of constant acceleration.

Step-by-Step SUVAT Problem-Solving Method

Use this method for most SUVAT physics questions.

Step 1: Choose a positive direction

For example:

• upward = positive
• downward = negative

or:

• right = positive
• left = negative

Step 2: Write the five variables

Create a list:

[
s,\quad u,\quad v,\quad a,\quad t
]

Fill in everything you know.

Step 3: Include signs

For example, if upward is positive:

[
a=-9.81,\text{m/s}^2
]

during free fall.

Step 4: Identify the unknown

Determine which quantity the question asks for.

Step 5: Identify the variable you do not have

Choose the SUVAT equation that contains your known quantities and unknown while excluding the unnecessary variable.

Step 6: Substitute and solve

Always use consistent SI units:

• displacement → m
• velocity → m/s
• acceleration → m/s²
• time → s

Understanding SUVAT Sign Conventions

Sign conventions are one of the most common sources of errors.

Suppose upward is positive.

For an object thrown upward:

[
u>0
]

At the highest point:

[
v=0
]

Gravity acts downward:

[
a=-9.81,\text{m/s}^2
]

If downward is instead selected as positive, then gravity becomes:

[
a=+9.81,\text{m/s}^2
]

Both choices are valid as long as the same convention is used consistently.

SUVAT and Velocity-Time Graphs

SUVAT equations can also be understood using velocity-time ((v-t)) graphs.

For constant acceleration:

Gradient

The gradient of a velocity-time graph is acceleration:

[
a=\frac{\Delta v}{\Delta t}
]

Area

The area under a velocity-time graph represents displacement:

[
s=\text{area under the }v-t\text{ graph}
]

Intercept

The velocity at (t=0) is the initial velocity (u).

This graphical interpretation provides another way to understand why the SUVAT equations work.

Worked Examples Using SUVAT Equations

Example 1: Stopping Distance

A car travels at (25,\text{m/s}) and brakes with a constant deceleration of (5,\text{m/s}^2). Find the stopping distance.

Given:

[
u=25,\text{m/s}
]

[
v=0
]

[
a=-5,\text{m/s}^2
]

We do not know time, so use:

[
v^2=u^2+2as
]

[
0=25^2+2(-5)s
]

[
0=625-10s
]

Therefore:

[
\boxed{s=62.5,\text{m}}
]

Example 2: Free Fall

A stone is dropped from rest and falls for (3.5) seconds. Find its displacement using (g=9.81,\text{m/s}^2).

Take downward as positive.

[
u=0
]

[
a=9.81,\text{m/s}^2
]

[
t=3.5,\text{s}
]

Use:

[
s=ut+\frac12at^2
]

[
s=0+\frac12(9.81)(3.5)^2
]

[
\boxed{s\approx60.1,\text{m}}
]

Example 3: Maximum Height

A ball is thrown vertically upward at (18,\text{m/s}). Find its maximum height.

At maximum height:

[
v=0
]

Take upward as positive:

[
u=18,\text{m/s}
]

[
a=-9.81,\text{m/s}^2
]

Use:

[
v^2=u^2+2as
]

[
0=18^2+2(-9.81)s
]

[
s=\frac{324}{19.62}
]

[
\boxed{s\approx16.5,\text{m}}
]

Example 4: Train Acceleration

A train starts from rest and travels (200) m in (10) seconds with constant acceleration. Find its acceleration and final velocity.

Given:

[
u=0,\quad s=200,\text{m},\quad t=10,\text{s}
]

Use:

[
s=ut+\frac12at^2
]

[
200=\frac12a(10)^2
]

[
200=50a
]

Therefore:

[
\boxed{a=4.0,\text{m/s}^2}
]

Now use:

[
v=u+at
]

[
v=0+(4)(10)
]

[
\boxed{v=40,\text{m/s}}
]

Example 5: Finding Time

A sprinter slows from (12,\text{m/s}) to (4,\text{m/s}) over a displacement of (32) m. Find the time taken.

Use:

[
s=\frac12(u+v)t
]

[
32=\frac12(12+4)t
]

[
32=8t
]

Therefore:

[
\boxed{t=4.0,\text{s}}
]

SUVAT in Projectile Motion

SUVAT equations can be used for projectile motion, but the horizontal and vertical components must be treated separately.

For a projectile launched with initial speed (u) at angle (\theta):

[
u_x=u\cos\theta
]

[
u_y=u\sin\theta
]

Ignoring air resistance:

Horizontal motion

[
a_x=0
]

Therefore:

[
x=u_xt
]

Vertical motion

[
a_y=-g
]

and SUVAT equations can be used vertically:

[
v_y=u_y-gt
]

[
y=u_yt-\frac12gt^2
]

This is the correct way to apply SUVAT to projectile motion.

Worked Projectile Example

A ball is launched at (20,\text{m/s}) at (30^\circ) above the horizontal. Ignore air resistance. Find its maximum height, time of flight, and horizontal range when it lands at the same height.

Step 1: Resolve the velocity

[
u_x=20\cos30^\circ\approx17.3,\text{m/s}
]

[
u_y=20\sin30^\circ=10,\text{m/s}
]

Step 2: Maximum height

At the top:

[
v_y=0
]

Use:

[
v_y^2=u_y^2+2a_y s
]

[
0=10^2+2(-9.8)H
]

[
H=\frac{100}{19.6}
]

[
\boxed{H\approx5.1,\text{m}}
]

Step 3: Time to maximum height

[
v_y=u_y-gt
]

[
0=10-9.8t
]

[
t_{\text{up}}\approx1.02,\text{s}
]

Therefore:

[
T\approx2.04,\text{s}
]

Step 4: Horizontal range

[
x=u_xT
]

[
x=17.3(2.04)
]

[
\boxed{x\approx35.3,\text{m}}
]

Vector SUVAT Equations

The standard SUVAT equations are generally taught for one-dimensional motion. However, they can be applied to vector components when motion is resolved into independent axes.

For projectile motion:

[
v_x=u_x+a_xt
]

and

[
v_y=u_y+a_yt
]

Similarly:

[
x=u_xt+\frac12a_xt^2
]

[
y=u_yt+\frac12a_yt^2
]

Thus, rather than treating SUVAT as one single three-dimensional vector formula, solve each independent component using the appropriate constant acceleration.

SUVAT on Inclined Planes

SUVAT can also be used on inclined planes after determining the acceleration along the slope.

For a frictionless incline at angle (\theta):

[
a=g\sin\theta
]

If friction is present:

[
a=g(\sin\theta-\mu\cos\theta)
]

for an object sliding down the slope under the stated assumptions.

Example

A block slides from rest down a frictionless (30^\circ) incline for (5) m. Find its final speed.

First:

[
a=g\sin30^\circ
]

[
a=9.8(0.5)=4.9,\text{m/s}^2
]

Now use:

[
v^2=u^2+2as
]

[
v^2=0+2(4.9)(5)
]

[
v^2=49
]

Therefore:

[
\boxed{v=7.0,\text{m/s}}
]

What If Acceleration Is Not Constant?

SUVAT equations have a strict limitation.

They apply only when:

[
\boxed{a=\text{constant}}
]

If acceleration varies with time, velocity, or position, the standard SUVAT equations cannot generally be applied over the entire interval.

For variable acceleration, use relationships such as:

[
a=\frac{dv}{dt}
]

and

[
v=\frac{ds}{dt}
]

which can be solved using calculus.

Examples where ordinary SUVAT may not apply include:

• variable acceleration
• significant air resistance
• changing acceleration in powered motion
• general circular motion
• motion involving different acceleration phases

The Split-Journey Rule

If acceleration changes during a journey, divide the motion into separate stages.

For example, if a ball is thrown upward and then falls downward:

Stage 1

Launch to maximum height.

Stage 2

Maximum height to the ground.

SUVAT can be applied separately to each stage because each stage can have a constant acceleration.

Do not automatically apply one SUVAT equation across a journey where the acceleration or direction conditions have changed.

SUVAT and Free Fall

Free-fall problems are among the most common applications of SUVAT.

Near Earth’s surface, ignoring air resistance:

[
g\approx9.81,\text{m/s}^2
]

or, in many school problems:

[
g\approx9.8,\text{m/s}^2
]

Some examination questions use:

[
g=10,\text{m/s}^2
]

Always use the value specified by the question.

If upward is positive:

[
a=-g
]

If downward is positive:

[
a=+g
]

The sign depends on your chosen coordinate system.

SUVAT Equations for Time

There is no single special “SUVAT equation for time.” Instead, time can be found by rearranging suitable equations.

For example:

[
v=u+at
]

gives:

[
\boxed{t=\frac{v-u}{a}}
]

And:

[
s=\frac12(u+v)t
]

gives:

[
\boxed{t=\frac{2s}{u+v}}
]

The appropriate equation depends on which variables are known.

Can SUVAT Be Used Without One of the Variables?

Yes. This is one of the main reasons there are five standard equations.

Without (u)

[
s=vt-\frac12at^2
]

Without (v)

[
s=ut+\frac12at^2
]

Without (s)

[
v=u+at
]

Without (a)

[
s=\frac12(u+v)t
]

Without (t)

[
v^2=u^2+2as
]

This is often the quickest way to select the correct formula in an exam.

SUVAT Calculator or Solver: How to Choose the Formula

A dedicated SUVAT calculator or solver normally requires you to enter known values from (s,u,v,a,t), identify the unknown, and calculate it using an appropriate equation.

You can perform the same process manually:

1. Write (s,u,v,a,t).
2. Enter the known quantities.
3. Mark the unknown.
4. Identify the missing variable.
5. Select the SUVAT equation that does not require the unnecessary variable.
6. Rearrange if required.
7. Check units and signs.

For example, if:

[
u=10,\text{m/s}
]

[
a=2,\text{m/s}^2
]

[
t=5,\text{s}
]

and the question asks for (v), use:

[
v=u+at
]

[
v=10+(2)(5)
]

[
\boxed{v=20,\text{m/s}}
]

A calculator can reduce arithmetic errors, but it cannot replace choosing the physically appropriate equation.

SUVAT for IB Physics, IGCSE, A-Level, and AP Physics

SUVAT equations are widely used in introductory mechanics curricula.

Students preparing for IB Physics, IGCSE Physics, A-Level Physics, or AP Physics should be comfortable with:

• identifying (s,u,v,a,t)
• applying sign conventions
• rearranging equations
• interpreting velocity-time graphs
• solving free-fall problems
• solving projectile components
• distinguishing displacement from distance
• recognizing when acceleration is constant
• selecting the equation with the required variables

Always follow the notation and value of (g) specified by your particular syllabus or examination question.

Common SUVAT Exam Mistakes

1. Treating distance as displacement

Remember:

[
s=\text{displacement}
]

not necessarily total distance.

2. Using the wrong sign for (g)

Choose your positive direction first.

3. Assuming negative acceleration always means slowing down

An acceleration vector can be negative while an object is moving in the negative direction and actually speeding up.

4. Using SUVAT when acceleration changes

SUVAT requires constant acceleration over the interval being analyzed.

5. Mixing units

Convert quantities into compatible SI units before substituting.

For example:

[
1,\text{km/h}=\frac{1}{3.6},\text{m/s}
]

6. Using one equation for a multi-stage journey

Split the journey whenever acceleration or the physical conditions change.