# Distance, speed and acceleration

You will be aware of the definitions of distance, speed and acceleration. These can also be written in derivative form:

You will also know about the SUVAT equations, but at university these can’t be assumed, so you need to be able to derive them.

You know that final speed is initial speed plus the change in speed. From the derivative forms above you also know that the change in speed is acceleration integrated with respect to time, so we can derive the first SUVAT equation:

In the same way, position is the initial position plus the change in position, which is velocity integrated with respect to time. We now know velocity from the first equation, so we substitute this in and solve:

We can now substitute

from the first equation into the second equation, and this rearranges to give another of the SUVAT equations:

The remaining SUVAT equation:

comes from the fact that distance traveled is equal to average speed times time, and average speed is half the initial speed plus the final speed.

SUVAT equations are used when an object is accelerating constantly. This means they can also be used when an object is falling due to gravity, even if it’s travelling with constant speed in the direction perpendicular to acceleration. This means they can be used for problems involving projectile motion.

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