What are the SUVAT equations in physics and kinematics?

The SUVAT equations are a set of five fundamental kinematic equations that describe the motion of an object moving with constant (uniform) linear acceleration. The acronym SUVAT stands for: s (displacement), u (initial velocity), v (final velocity), a (acceleration), and t (time).

What are the five core SUVAT formulas?

The 5 standard SUVAT equations are: (1) v = u + at, (2) s = [(u + v) / 2]·t, (3) s = ut + ½at², (4) s = vt - ½at², and (5) v² = u² + 2as. Each formula omits exactly one of the five kinematic variables.

How many SUVAT variables do you need to solve for the remaining unknowns?

You need exactly three known variables to determine the remaining two unknowns. Because there are 5 variables in total, giving any 3 allows you to select the appropriate equations to solve for the other 2 variables algebraically.

How do you choose the right SUVAT equation for a physics problem?

Identify the three given variables and the target unknown variable you want to find. Then identify the one variable that is neither given nor required (the omitted variable), and choose the single SUVAT equation that does not contain that omitted variable.

What is the difference between displacement (s) and distance in SUVAT?

Displacement (s) is a vector quantity representing the straight-line distance and direction from the starting point to the ending point (can be positive, negative, or zero). Distance is a scalar quantity measuring the total path length traveled (always non-negative).

Can the SUVAT equations be used when acceleration changes over time?

No. SUVAT equations are strictly valid only for uniform, constant acceleration (a = constant). If acceleration varies with time or position (e.g. aerodynamic drag, rocket propellant burn-off, simple harmonic motion), calculus integration (v = ∫a dt and s = ∫v dt) must be used instead.

How do signs (+ and -) work in SUVAT kinematics problems?

You must define a consistent reference direction (typically upward or to the right as positive). If upward is positive, downward gravitational acceleration is negative (a = -9.81 m/s²), an upward initial velocity is positive (u > 0), and landing below the launch point gives a negative displacement (s < 0).

Why does solving for time (t) in s = ut + ½at² sometimes give two answers?

The equation s = ut + ½at² is quadratic in time (½at² + ut - s = 0). When a projectile is launched upward and returns to ground, two positive roots represent the two distinct moments the object passes through a specific elevation (once while ascending, and once while descending).

How do you use SUVAT equations for vertical free fall and projectile motion?

Set acceleration to gravitational acceleration (a = -9.81 m/s² on Earth). At maximum apex height, vertical velocity reaches zero (v = 0). For an object dropped from rest, initial velocity is zero (u = 0), simplifying displacement to s = ½gt² and velocity to v = gt.

How do engineers, physicists, and students use the SUVAT Calculator?

High school and university AP/A-Level physics students use the SUVAT Calculator to solve coursework problem sets, automotive engineers calculate vehicle stopping distances and crash deceleration rates, and aerospace ballistics analysts model multi-stage rocket acceleration profiles.

⚡ 100% Free • All 5 Kinematic Equations & Constant Acceleration Solver

SUVAT Calculator

Solve all 5 constant acceleration kinematics equations with the free in-browser SUVAT Calculator. Input any 3 variables (displacement s, initial velocity u, final velocity v, acceleration a, time t) to calculate the remaining 2 variables with step-by-step mathematical proofs.

Kinematics SUVAT Presets:

SUVAT Input Status: 3 of 3 Knowns Entered • Ready

Enter any 3 known variables and leave the remaining 2 unknown fields blank.

1. Displacement (s) GIVEN

2. Initial Velocity (u) GIVEN

3. Final Velocity (v) GIVEN

4. Acceleration (a) GIVEN

5. Time Duration (t) UNKNOWN

Computed Unknowns

s = 75.00 m • t = 5.00 s

Average Velocity: 15.00 m/s • Deceleration Stopping Distance: 75.00 m

Displacement (s) CALC

75.00 m

246.06 ft

Initial Speed (u) GIVEN

30.00 m/s

108.00 km/h

Final Speed (v) GIVEN

0.00 m/s

0.00 km/h

Acceleration (a) GIVEN

-6.00 m/s²

-0.61 g

Time (t) CALC

5.00 s

5,000 ms

Average Speed DERIVED

15.00 m/s

54.00 km/h

Step-by-Step SUVAT Algebraic Derivation

The Foundations of SUVAT & Constant Acceleration Kinematics

In classical Newtonian mechanics, the SUVAT equations (also known as the kinematic equations of uniform acceleration) describe the linear motion of a body experiencing constant acceleration. The acronym represents the five fundamental kinematic quantities:

s = Displacement

Net change in position from origin (vector in meters)

u = Initial Velocity

Starting velocity at time $t = 0$ (vector in m/s)

v = Final Velocity

Velocity at elapsed time $t$ (vector in m/s)

a = Acceleration

Constant rate of velocity change (vector in m/s²)

t = Time Duration

Elapsed time interval (scalar in seconds)

The 5 Fundamental SUVAT Formulas & Omitted Variables Matrix

Every SUVAT formula links four of the five variables, intentionally omitting exactly one:

Equation Number	SUVAT Kinematic Formula	Omitted Variable	Primary Problem Use Case
Equation 1	$$v = u + at$$	Displacement ($s$)	Finding final speed when distance is unknown
Equation 2	$$s = \frac{u + v}{2} \cdot t$$	Acceleration ($a$)	Average velocity displacement problems
Equation 3	$$s = ut + \frac{1}{2}at^2$$	Final Velocity ($v$)	Free fall drops and vehicle acceleration distance
Equation 4	$$s = vt - \frac{1}{2}at^2$$	Initial Velocity ($u$)	Braking problems where final stop speed is known
Equation 5	$$v^2 = u^2 + 2as$$	Time Duration ($t$)	Timeless crash impact and stopping distance calculations

How to Use the SUVAT Calculator

1 Identify Your 3 Known Quantities

Determine which three variables from $s, u, v, a, t$ are provided in your kinematics problem statement.

2 Enter Values & Set Units

Input the numbers into the corresponding fields and select your preferred units. Leave the two unknown fields completely empty.

3 Ensure Consistent Vector Signs

Use negative signs for decelerating vehicles ($a < 0$) or downward gravity ($a = -9.81\text{ m/s}^2$).

4 Review Solved Values & Proofs

The solver automatically identifies the necessary formulas, calculates both missing variables, and renders the algebraic proof.

Vector Sign Conventions: Defining Direction & Deceleration

Because $s, u, v,$ and $a$ are vector quantities, choosing a reference coordinate system is critical:

Horizontal Linear Motion

If forward motion is positive ($+x$), a car accelerating forward has $u > 0$ and $a > 0$. When braking, acceleration opposes velocity, requiring negative acceleration ($a < 0$).

Vertical Free Fall & Projectiles

If upward is chosen as positive ($+y$), gravity points downward ($a = -9.81\text{ m/s}^2$). An upward toss has $u > 0$, reaches $v = 0$ at apex, and lands with $v < 0$ and $s = 0$.

Quadratic Time Solving: Resolving Roots in $s = ut + \frac{1}{2}at^2$

When solving for time duration ($t$) given displacement ($s$), initial velocity ($u$), and acceleration ($a$), the formula rearranges into a standard quadratic equation:

$$\frac{1}{2}at^2 + ut - s = 0 \implies t = \frac{-u \pm \sqrt{u^2 + 2as}}{a}$$

In physical kinematics, negative time roots ($t < 0$) represent mathematical extrapolation before motion began and are discarded, whereas two positive roots correspond to passing an elevation on both ascent and descent.

Key Features of the SUVAT Calculator

All 10 Combinations Supported

Solves any 3-variable combination from the 5 SUVAT kinematic parameters automatically.

Live Variable Status Badges

Instantly tags variables as GIVEN or CALCULATED across the dashboard grid.

Step-by-Step KaTeX Math

Renders clean mathematical proofs showing algebraic rearrangement and exact numerical substitution.

Comprehensive Unit Engine

Inputs and outputs across meters, feet, km, miles, m/s, km/h, mph, ft/s, m/s², and g-force.

Deceleration & Stopping Distances

Models automotive brake testing, runway takeoffs, and dragster acceleration intervals.

100% In-Browser & Private

All kinematic calculations execute locally with zero latency and complete data privacy.

Automotive Braking & Highway Stopping Distance Kinematics

How transportation safety engineers compute total vehicle stopping distance:

Total stopping distance equals Reaction Distance plus Braking Distance:

$$s_{\text{total}} = (u \cdot t_{\text{reaction}}) + \frac{u^2}{2|a_{\text{brake}}|}$$

For a car traveling at $108\text{ km/h}$ ($30\text{ m/s}$) with a $0.75\text{ s}$ driver reaction time and maximum dry-pavement braking deceleration of $a = -6\text{ m/s}^2$, total stopping distance is $s = (30 \times 0.75) + \frac{900}{12} = 22.5\text{ m} + 75.0\text{ m} = \mathbf{97.5\text{ meters}}$.

Problems This SUVAT Calculator Solves

Eliminates Algebraic Substitution Errors

Selects the correct kinematic formula from the 5 SUVAT equations and solves simultaneous systems without manual algebra.

Handles Multi-Unit Conversions Instantly

Seamlessly bridges metric and imperial velocity, acceleration, and distance units without unit conversion errors.

Calculus Foundations: Deriving SUVAT from First Principles ($a \to v \to s$)

How definite integrals of constant acceleration yield the kinematic formulas:

$$\int_{u}^{v} dv = \int_{0}^{t} a\,dt \implies v = u + at \qquad \int_{0}^{s} ds = \int_{0}^{t} (u + at)\,dt \implies s = ut + \frac{1}{2}at^2$$

Eliminating time $t = \frac{v - u}{a}$ and substituting into the displacement integral establishes the timeless third law of kinematics: $v^2 = u^2 + 2as$.

Aerospace & Defense: Multi-Stage Rocket Acceleration Modeling

How propulsion engineers model piecewise linear launch trajectories:

Stage 1 Booster Burn

With $u_0 = 0$ and net booster thrust acceleration $a_1 = 25\text{ m/s}^2$ over $t_1 = 60\text{ s}$, burnout velocity reaches $v_1 = 1,500\text{ m/s}$ at an altitude of $s_1 = 45\text{ km}$.

Stage 2 Vacuum Insertion

Stage 1 burnout velocity becomes the initial velocity for Stage 2 ($u_2 = v_1 = 1,500\text{ m/s}$), accelerating continuously toward orbital velocity ($7,800\text{ m/s}$).

High-Speed Railway Kinematics & Passenger G-Force Thresholds

How bullet train engineers balance rapid acceleration with passenger safety:

To maintain passenger comfort without seatbelts, high-speed rail systems (e.g. Japanese Shinkansen, French TGV) strictly cap linear acceleration to $a \le 0.70\text{ m/s}^2$ ($0.071\,g$). Accelerating from rest to $320\text{ km/h}$ ($88.89\text{ m/s}$) requires $t = 88.89 / 0.70 = \mathbf{127.0\text{ seconds}}$ and a track runway displacement of $s = \frac{1}{2}(0.70)(127)^2 = \mathbf{5.64\text{ kilometers}}$.

Aircraft Carrier Catapults & Arresting Wire Kinematics (EMALS)

How naval aviation catapults launch 30-ton fighter jets off short flight decks:

Catapult Launch Stroke

Accelerating from $u = 0$ to takeoff speed $v = 72.2\text{ m/s}$ ($260\text{ km/h}$) over an $s = 90\text{-meter}$ track requires:

$$a = \frac{v^2}{2s} = \frac{(72.2)^2}{180} = \mathbf{28.96\text{ m/s}^2} \quad (2.95\,g) \quad [t = 2.49\text{ s}]$$

Trap Wire Landing Deceleration

Arresting an aircraft from $v_0 = 65\text{ m/s}$ to full stop over $s = 95\text{ meters}$ generates $a = -22.2\text{ m/s}^2$ ($-2.27\,g$) in $t = 2.92\text{ seconds}$.

Roller Coaster Engineering: Gravity Drops & Clothoid Loop Inversions

How amusement park designers use SUVAT conservation equations:

A coaster dropping from a $s = 60\text{-meter}$ lift hill reaches a bottom valley speed of $v = \sqrt{2gs} = \sqrt{2(9.81)(60)} = \mathbf{34.31\text{ m/s}}$ ($123.5\text{ km/h}$). Designers shape loops with teardrop clothoid geometry to control centripetal acceleration ($a_c = v^2/R$), keeping passenger loads safely within $+4.0\,g$ to $-1.0\,g$.

Forensic Accident Reconstruction: Skid Mark Velocity Formula

How crash investigators determine pre-braking vehicle speed:

Given the road friction coefficient ($\mu$) and skid mark distance ($s$), the vehicle's speed before wheel lockup is solved via $v^2 = u^2 - 2(\mu g)s = 0$:

$$u = \sqrt{2 \mu g s}$$

A $45\text{-meter}$ skid on dry asphalt ($\mu = 0.75$) proves an initial speed of $u = \sqrt{2(0.75)(9.81)(45)} = \mathbf{25.73\text{ m/s}}$ ($92.6\text{ km/h} \approx 57.6\text{ mph}$).

Frequently Asked Questions

Comprehensive answers to common questions about SUVAT equations, constant acceleration formulas, sign conventions, and kinematics problems.