# 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:

1. $\hspace{10pt}S=UT+\frac{1}{2}AT^2$
2. $\hspace{10pt}S=\left(\frac{U+V}{2}\right)T$
3. $\hspace{10pt}V=U+AT$
4. $\hspace{10pt}V^2=U^2+2AS$
Variable Description SI unit
S displacement $m$
U initial velocity $m/s$
V final velocity $m/s$
A acceleration $m/s^2$
T total time $s$

## Derivation of the SUVAT Equations

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

$\text{acceleration} = \frac{\text{change in velocity}}{\text{change in time}}\hspace{3pt}\text{or}\hspace{3pt}A=\frac{V-U}{T}$

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:

$V=U+AT$

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 $S=\frac{1}{2}(U+V)T$ comes from the fact that acceleration is constant. In this case, $\frac{1}{2}(U+V)$ 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 $T=\frac{V-U}{A}$ which can be substituted into equation 2:
$S=\left(\frac{U+V}{2}\right)\left(\frac{V-U}{A}\right)$ and rearranged gives $V^2=U^2+2AS$
• Substitute the expression for V in SUVAT Equation 1 directly into SUVAT Equation 2:
$\begin{array}{l}S&=&\left(\frac{U+U+AT}{2}\right)T\\&=&UT+\frac{1}{2}AT^2\end{array}$
• SUVAT Equation 1 can be rearranged to make U the subject so that $U=V-AT$. Substitute this into equation 4 to give
$\begin{array}{l}S&=&\left(V-AT\right)T+\frac{1}{2}AT^2\\&=&VT-\frac{1}{2}AT^2\end{array}$

See the Examples for use of the SUVAT equations and an example derivation question.  When attempting examples for yourself, make sure that the dimensions are consistent. In other words, you should ensure that you are using the same SI unit for all measurements. Accordingly, this may require a conversion. See Motion under Gravity for more examples using the SUVAT equations. Dr Nat

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