# The SUVAT Equations and their derivation

The SUVAT Equations describe motion in a given direction when **ACCELERATION IS CONSTANT**. The SUVAT Equations that are given in the Formula Booklet are:

Variable | Description | SI unit |
---|---|---|

S | displacement | |

U | initial velocity | |

V | final velocity | |

A | acceleration | |

T | total time |

## Derivation of the SUVAT Equations

In general, for motion in a straight line with constant acceleration:

where V is the final velocity, U is the initial velocity and T is the total time taken. Rearranging gives the equation in an alternative form:

This equation is one of the **SUVAT **equations. They are named so since they involve displacement (S), initial velocity (U), final velocity (V), acceleration (A) and time (T) for motion in a straight line with constant acceleration. Note that in order to use the SUVAT equations, we must define a reference point with regards to the displacement. We must also specify the direction of positive and negative speed. Speed cannot be negative but acceleration can. If an object is slowing down rather than speeding up, acceleration is negative.

The second SUVAT equation comes from the fact that acceleration is constant. In this case, is the average speed throughout the duration of travel. Multiplying this by T will give the total distance. This is because distance is speed multiplied by time when acceleration is constant.

The first two SUVAT equations can be used to derive the remaining SUVAT equations:

- SUVAT Equation 1 can be rearranged to make T the subject so that which can be substituted into equation 2:

and rearranged gives - Substitute the expression for V in SUVAT Equation 1 directly into SUVAT Equation 2:
- SUVAT Equation 1 can be rearranged to make U the subject so that . Substitute this into equation 4 to give

YOU ARE EXPECTED TO KNOW HOW TO DERIVE THE SUVAT EQUATIONS.

See the Examples below for different applications of the SUVAT equations. When attempting examples for yourself, make sure that the dimensions are consistent. In other words, you should ensure that you are using the same units for all measurements. Accordingly, this may require a conversion as in Examples 3 and 4. You should also see Motion under Gravity for more examples using the SUVAT equations.

## Examples

## Videos

This video will show you how to tackle a basic SUVAT equations question by first setting up the equations then showing you how to select which one to use. This particular question asks students how to find how long it takes for a ball thrown in the air takes to reach the ground under certain conditions. Initial speed is given and gravity is set to 10m/s^2. There were two exam questions on this paper using SUVAT equations – SUVAT Equation 2 will show you how to answer the second.

This video will show you how to tackle a more involved SUVAT equations question. There are several parts to the question – some use SUVAT in the several parts, some use a velocity-time graph and the final asks the student to suggest an improvement to the model. There were two exam questions on this paper using SUVAT equations – SUVAT Equation 1 will show you how to answer the other one which was Question 6 on this paper. This question is a little more basic and sets up the SUVAT equations before showing you how to select which one to use to find the time taken for a ball thrown in the air to reach the ground.