## Motion in a straight line with constant acceleration

Recall that acceleration at a given time measures the instantaneous change in velocity – see Quantities & Units in Mechanics. For motion in a straight line with constant acceleration, the speed along the line changes by the same amount every second. Hence, if speed is measured in metres per second (m/s) then acceleration is measured in metres per second per second (m/s/s). For example, consider a ball rolling along a flat surface:

If the ball rolls with speed 4 m/s and accelerates at a constant rate of 2 m/s/s, the ball will have speed 10 m/s after 3 seconds.

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

$\text{acceleration} = \frac{\text{change in velocity}}{\text{change in time}}\hspace{30pt}\text{or}\hspace{30pt}A=\frac{V-U}{T}$

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:

$V=U+AT$

Variable/Constant Description SI unit
S displacement m (metres)
U initial velocity m/s (metres per second)
V final velocity m/s (metres per second)
A acceleration m/s/s (metres per second per second)
T total time s (seconds)

These two equations are two of the SUVAT equations 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.

Click here for the other SUVAT equations.

Note that in order to use these equations, a reference point with regards to the displacement must be defined. The direction of positive and negative speed must also be determined. In the example with the rolling ball above, left is the positive direction. Speed cannot be negative but acceleration can. Negative acceleration indicates that something is slowing down rather than speeding up. (TBC)

When attempting examples for yourself, make sure that the dimensions are consistent, i.e. you are using the same SI unit for all measurements (which may require a conversion) and the arithmetic of dimensions is consistent. The latter is known as dimensional analysis. (TBC)

## Distance-Time and Speed-Time Graphs

The gradient on a distance-time graph gives you the instantaneous measure of speed.
The gradient on a speed-time graph gives you the instantaneous measure of acceleration. The total area beneath the speed-time graph gives you the total distance measured. (TBC)