SUVAT Equations Calculator

General Physics

This tool solves the equations of motion for a body moving with constant acceleration. Enter any three of the five variables (s, u, v, a, t) to calculate the other two.

Practical Examples

See how the SUVAT calculator is applied in different scenarios. Click on an example to load the data.

Object in Free Fall

Free Fall

Calculate the final velocity and time taken for an object dropped from a height of 150 meters.

Displacement (s): 150 m

Initial Velocity (u): 0 m/s

Acceleration (a): 9.81 m/s²

Car Accelerating

Vehicle Acceleration

A car accelerates from rest to 27 m/s (approx 100 km/h) over a distance of 250 meters. Find its acceleration and the time taken.

Initial Velocity (u): 0 m/s

Final Velocity (v): 27 m/s

Displacement (s): 250 m

Vehicle Braking

Braking

A car traveling at 30 m/s brakes to a stop in 4 seconds. Calculate the braking acceleration and the distance traveled.

Initial Velocity (u): 30 m/s

Final Velocity (v): 0 m/s

Time (t): 4 s

Vertical Projectile

Projectile Motion

A ball is thrown upwards with an initial velocity of 20 m/s. Find the maximum height it reaches and the time it takes to get there.

Initial Velocity (u): 20 m/s

Final Velocity (v): 0 m/s

Acceleration (a): -9.81 m/s²

Other Titles
Understanding the SUVAT Equations: A Comprehensive Guide
An in-depth look at the fundamental equations of motion for objects with constant acceleration.

What are the SUVAT Equations?

  • Defining the Five Kinematic Variables
  • The Core Assumption: Constant Acceleration
  • The Five SUVAT Equations
The SUVAT equations are a set of five formulas used in kinematics to describe the motion of an object under constant, uniform acceleration. The acronym 'SUVAT' itself is derived from the five variables these equations relate: Displacement (s), Initial Velocity (u), Final Velocity (v), Acceleration (a), and Time (t). By knowing any three of these variables, one can determine the other two, making these equations incredibly powerful for solving a wide range of physics problems.
The Five Variables
s (Displacement): The vector quantity representing the change in the object's position. It's not the same as distance, which is a scalar.
u (Initial Velocity): The vector quantity for the object's velocity at the start of the observed time period (t=0).
v (Final Velocity): The vector quantity for the object's velocity at the end of the observed time period.
a (Acceleration): The vector quantity for the rate of change of velocity. In SUVAT, this must be constant.
t (Time): The scalar quantity for the duration over which the motion is observed.
The Five Equations
v = u + at
s = ut + ½at²
v² = u² + 2as
s = vt - ½at²
s = ½(u + v)t

Step-by-Step Guide to Using the SUVAT Calculator

  • Identifying Known Variables
  • Inputting Values Correctly
  • Interpreting the Results
Using this calculator is a straightforward process designed to give you accurate results quickly. Follow these steps to solve your kinematics problem.
Step 1: Identify Your Knowns and Unknowns
First, read your physics problem carefully and identify the three quantities you have been given. Also, determine which two quantities you need to find. For example, a problem might state a car's initial velocity, its acceleration, and the time it accelerates for. Here, your knowns are u, a, and t, and your unknowns are s and v.
Step 2: Enter the Three Known Values
Enter the three known values into their corresponding input fields in the calculator. Leave the fields for the two unknown variables blank. The calculator is designed to automatically detect which variables are provided and which need to be calculated.
Step 3: Calculate and Interpret
Click the 'Calculate' button. The tool will instantly compute the two unknown variables and display them in the 'Results' section. Ensure you understand the units of the results, which will be consistent with the units of your inputs. For example, if you input velocity in m/s and time in s, the displacement will be in meters.

Real-World Applications of SUVAT

  • Automotive Engineering and Road Safety
  • Sports Science and Biomechanics
  • Celestial Mechanics and Aerospace
The SUVAT equations are not just for textbook problems; they have numerous applications in the real world for analyzing and predicting motion.
Automotive Engineering
Engineers use these principles to design vehicles, calculate braking distances, and improve safety features. For instance, determining the stopping distance of a car at a certain speed is a direct application of v² = u² + 2as.
Sports Science
In biomechanics, SUVAT helps analyze athlete performance. For example, calculating the maximum height of a high jumper's center of mass or the velocity of a shot put at release involves these equations.
Aerospace and Projectile Motion
Calculating the trajectory of a projectile, from a simple thrown ball to a rocket launch (in its initial phase), relies heavily on SUVAT. They help determine flight time, maximum altitude, and range, assuming constant acceleration (like gravity) and negligible air resistance.

Mathematical Derivation and Examples

  • Deriving v = u + at from the Definition of Acceleration
  • Deriving s = ut + ½at² from Velocity-Time Graphs
  • Worked Example: Calculating Braking Distance
Understanding where the SUVAT equations come from can provide a deeper insight into their use. They are derived from the basic definitions of velocity and acceleration.
Derivation of v = u + at
Acceleration (a) is defined as the rate of change of velocity. Mathematically, a = (v - u) / t. Rearranging this simple definition directly gives the first SUVAT equation: v = u + at.
Derivation via Velocity-Time Graph
The displacement (s) is the area under a velocity-time graph. For an object with constant acceleration, this graph is a straight line, and the area underneath is a trapezoid. The area of a trapezoid is given by ½(sum of parallel sides) × height. In this context, the parallel sides are u and v, and the height is t. This gives s = ½(u + v)t. By substituting v = u + at into this equation, we can derive s = ut + ½at².

Worked Example:

  • Problem: A car traveling at 20 m/s applies its brakes, producing a deceleration of 5 m/s². How far does it travel before stopping?
  • 1. Knowns: u = 20 m/s, v = 0 m/s (since it stops), a = -5 m/s².
  • 2. Unknown: s.
  • 3. Equation: Use v² = u² + 2as.
  • 4. Solve: 0² = 20² + 2(-5)s -> 0 = 400 - 10s -> 10s = 400 -> s = 40 meters.

Common Misconceptions and Key Considerations

  • Constant vs. Variable Acceleration
  • Vector Nature of s, u, v, and a
  • The Role of Air Resistance
Constant Acceleration is Key
The most critical limitation of the SUVAT equations is that they are only valid when acceleration is constant. If acceleration changes over time, more advanced techniques like calculus are required.
Direction Matters (Vectors)
Displacement, velocity, and acceleration are vector quantities. This means they have both magnitude and direction. In one-dimensional problems, direction is handled by assigning positive or negative signs to the values. For example, if 'up' is positive, the acceleration due to gravity (g) is negative (-9.81 m/s²). Be consistent with your sign convention throughout a problem.
Ignoring Air Resistance
Most introductory physics problems using SUVAT assume that air resistance and friction are negligible. In many real-world scenarios, these forces are significant and can cause acceleration to be non-uniform, which would make the SUVAT equations an approximation rather than an exact description.