What does SUVAT stand for?

SUVAT is an acronym for the five kinematic variables used in the equations of motion for uniform acceleration: s = displacement (metres), u = initial velocity (m/s), v = final velocity (m/s), a = acceleration (m/s²), t = time (seconds). The five SUVAT equations relate these variables for objects moving with constant (uniform) acceleration.

When can I use SUVAT equations?

SUVAT equations apply when acceleration is constant (uniform). This includes free fall near Earth's surface (constant g = 9.81 m/s²), cars accelerating uniformly, objects on inclined planes (constant gravitational component), and any other scenario where net force — and therefore acceleration — does not change with time. They do NOT apply to circular motion, oscillations, or variable-force situations.

How do I choose which SUVAT equation to use?

Each of the five SUVAT equations omits one variable. Identify which variable you do NOT know and do NOT need, then use the equation that omits it. For example: if you know u, a, t and want v and s, use v = u + at (omits s) and s = ut + ½at² (omits v). This calculator automatically selects the correct equation based on which three variables you provide.

Why do I need three variables to use SUVAT?

There are five variables (s, u, v, a, t) and the equations describe relationships among them under constant acceleration. Each equation involves four of the five variables. Knowing three variables gives you two equations with two unknowns, which is uniquely solvable. With only two known variables, you have more unknowns than equations and cannot find a unique solution.

What sign convention should I use in SUVAT?

Choose a positive direction and stick with it. Typically: upward is positive for vertical motion, and rightward (or the direction of initial motion) is positive for horizontal motion. Acceleration due to gravity is a = −9.81 m/s² if upward is positive (deceleration when moving up, acceleration when moving down). Displacement can be negative if the object ends up behind its starting point.

How is SUVAT different from calculus kinematics?

SUVAT equations are algebraic shortcuts derived by integrating the equations of motion under the special case of constant acceleration. Calculus kinematics (integration and differentiation of position, velocity, acceleration functions) applies to any acceleration — constant or variable. SUVAT is a subset: it gives the same answers as calculus when a is constant, but calculus is required when a varies with time.

Can SUVAT solve projectile motion?

Yes. For projectile motion, apply SUVAT separately to the horizontal and vertical components. Horizontal: a = 0, so s = ut gives horizontal range. Vertical: a = −9.81 m/s² (taking up as positive), apply the five SUVAT equations to find maximum height, time of flight, and vertical velocity at any point. This decomposition works because horizontal and vertical motions are independent.

What is the difference between speed and velocity in SUVAT?

Speed is the magnitude of velocity — it is always positive. Velocity is speed with direction — it can be positive or negative depending on your sign convention. In SUVAT, u and v are velocities (signed), not speeds. A ball thrown upward at 20 m/s has u = +20 m/s. At its peak, v = 0. After the peak, v is negative (moving downward). The magnitude |v| is the speed, which is always ≥ 0.

SUVAT Calculator – Equations of Motion Solver

Auto-fill a = 9.81 m/s² (free fall / gravity) Uncheck to enter custom a

Presets:

s Displacement

u Initial Velocity

v Final Velocity

a Acceleration

t Time (≥ 0)

Enter any 3 variables to solve for the other 2.

What Is SUVAT?

SUVAT refers to the five variables in the classical equations of motion for uniformly accelerated linear motion. The equations were derived by Galileo and formalised using Newton's laws. They are the backbone of GCSE and A-level physics kinematics.

s

Displacement (m)

u

Initial velocity (m/s)

v

Final velocity (m/s)

a

Acceleration (m/s²)

t

Time (s)

The Five Equations of Motion

#	Equation	Omits	Derivation hint
1	v = u + at	s	Definition of acceleration: a = Δv/t
2	s = ut + ½at²	v	Substitute v=u+at into s=½(u+v)t
3	v² = u² + 2as	t	Eliminate t between equations 1 and 2
4	s = ½(u+v)t	a	Average velocity × time
5	s = vt − ½at²	u	Substitute u=v−at into s=ut+½at²

How to Choose the Right Equation

Each equation omits one variable. The strategy: identify the variable you don't know and don't need, then use the equation that leaves it out.

Know u, a, t — want v: v = u + at (omits s)

Know u, a, t — want s: s = ut + ½at² (omits v)

Know u, v, a — want s: v² = u² + 2as (omits t)

Know u, v, t — want s: s = ½(u+v)t (omits a)

Know v, a, t — want s: s = vt − ½at² (omits u)

Sign Conventions and Vectors

SUVAT variables are vector quantities (except t). Choose a positive direction and be consistent throughout a problem.

Vertical motion (up = positive)

a = −9.81 m/s² (gravity downward)
Ball thrown up: u > 0, at peak v = 0, then v < 0
s > 0 means object is above start; s < 0 means below

Horizontal motion (right = positive)

Decelerating: a is negative (opposes motion)
s negative means object moved left of start
t is always positive (time cannot be negative)

Worked Examples (GCSE/A-level style)

Example 1 — Free fall for 3 s (u=0, a=9.81, t=3)

Known: u=0, a=9.81, t=3 | missing: s, v

Eq 1 (v=u+at): v = 0 + 9.81×3 = 29.43 m/s

Eq 2 (s=ut+½at²): s = 0×3 + ½×9.81×9 = 44.14 m

Answer: v = 29.43 m/s, s = 44.14 m

Example 2 — Car braking (u=30 m/s, v=0, s=45 m)

Known: u=30, v=0, s=45 | missing: a, t

Eq 3 (v²=u²+2as): 0 = 900 + 2a×45 → a = −10 m/s²

Eq 4 (s=½(u+v)t): 45 = ½×30×t → t = 3 s

Answer: a = −10 m/s², t = 3 s

Example 3 — Stone thrown up (u=20 m/s, a=−9.81, v=0)

Known: u=20, a=−9.81, v=0 | missing: s, t

Eq 1 (v=u+at): 0 = 20 − 9.81t → t = 2.039 s

Eq 3 (v²=u²+2as): 0 = 400 + 2×(−9.81)×s → s = 20.39 m

Answer: t = 2.04 s, s = 20.39 m (max height)

Example 4 — Train accelerating (u=10, v=40, t=120 s)

Known: u=10, v=40, t=120 | missing: a, s

Eq 1 (v=u+at): 40 = 10 + a×120 → a = 0.25 m/s²

Eq 4 (s=½(u+v)t): s = ½×(10+40)×120 = 3,000 m

Answer: a = 0.25 m/s², s = 3,000 m (3 km)

Limitations of SUVAT

SUVAT equations are powerful but limited to specific conditions:

✗Requires constant (uniform) acceleration throughout the motion
✗Does not apply to circular motion (direction of acceleration continuously changes)
✗Not valid for oscillatory or wave motion where acceleration varies
✗Cannot account for air resistance which produces velocity-dependent force
✓Valid for free fall (near surface), uniform braking/acceleration, motion on inclined planes

Frequently Asked Questions

SUVAT is an acronym for the five kinematic variables: s = displacement (m), u = initial velocity (m/s), v = final velocity (m/s), a = acceleration (m/s²), t = time (s). The five SUVAT equations relate these variables for objects moving with constant (uniform) acceleration.

SUVAT equations apply when acceleration is constant (uniform). This includes free fall near Earth's surface (constant g = 9.81 m/s²), cars accelerating uniformly, objects on inclined planes, and any scenario where net force — and therefore acceleration — does not change with time. They do NOT apply to circular motion, oscillations, or variable-force situations.

Each SUVAT equation omits one variable. Identify which variable you do NOT know and do NOT need, then use the equation that omits it. For example: if you know u, a, t and want v, use v = u + at (omits s). This calculator automatically selects the correct equation based on which three variables you provide.

There are five variables and each SUVAT equation involves four of them. Knowing three variables gives you sufficient information to find the remaining two, as each equation relates four variables and you have three already known. With only two known variables, you have more unknowns than equations, giving infinitely many solutions.

Choose a positive direction and be consistent. For vertical motion taking up as positive: a = −9.81 m/s². A ball thrown upward has u > 0; at the peak v = 0; descending v < 0. Displacement s can be negative if the object ends below its starting point. Time t is always positive.

SUVAT equations are algebraic shortcuts derived by integrating equations of motion under constant acceleration. Calculus kinematics applies to any acceleration — constant or variable. SUVAT gives the same answers as calculus when a is constant, but calculus is required when a varies with time (e.g., drag forces, spring forces).

Yes. For projectile motion, apply SUVAT separately to the horizontal (a = 0, s = ut) and vertical (a = −9.81 m/s²) components. This works because horizontal and vertical motions are independent. The horizontal and vertical analyses share the same time t, which links them together.

Speed is the magnitude of velocity and is always positive. Velocity is a vector — it can be positive or negative depending on the chosen direction. In SUVAT, u and v are velocities (signed). A ball thrown upward at 20 m/s has u = +20 m/s. At its peak v = 0. After the peak v is negative (moving downward). The magnitude |v| is the speed.

SUVAT Calculator (Equations of Motion)