# Velocity of an object which is decelerating

This might sound like a rather strange question, but I am slightly confused about the velocity of an object which is decelerating. Is the velocity just 0? Because that's what it is saying in this exercise question I found:

A cyclist starts from rest and accelerates at 1 m/s^2 for 20 seconds. He then travels at a constant speed for 1 minute and finally decelerates a 2m/s^2 until he stops. Find his maximum speed and the total distance he covered.

I have already found the first two parts, but not the thrird part (finally decelerates at 2m/s^2 until he stops).

I am using the formula

$$v^2 = u^2 + 2as$$

I rearranged it to make $s$ the subject. However, I don't know what the velocity is. Is it just $0$? If so, why?

• When you start doing such problems it might help you if you sketch velocity against time graphs. The gradient of such a graph is the acceleration and the area under the graph is the displacement. In this case your graph will have a straight line from the origin with positive gradient, then a plateau region (velocity constant) and then a straight line with negative gradient (decelerates = negative acceleration) ending up with a velocity of zero. – Farcher Sep 1 '17 at 9:26

The velocity isn't $0$ however it is changing at a rate of $$\frac{ds}{dt}$$If you draw out a velocity time graph then the area under the graph will be equal to the distance. As the graph is linear this should be relatively easy to do, and it will be in the shape of a trapezium. For non-linear systems i.e $$\frac{d^2s}{dt^2}\neq\mathrm{constant}$$you would need to know how the acceleration is changing the calculate the value for the distance.
If you still want to do the question in the way that you started out by using $$v^2=u^2+2as$$then you need to split it up into three sections, the start with positive acceleration, the middle with no acceleration and the end with negative acceleration. This is because the SUVAT equations only work for constant acceleration situations. In this scenario, $v$ is the final speed, and $u$ is the initial speed. As you know what $u$ is you can use the equation $$v=u+at$$ to calculate a value for the final speed.