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Part one of the question is "calculate the acceleration for the first 400m" which went just fine:

x=ut+(1/2)at^2

400=180.5a

a=2.22 m/s

Now the second part of the question asks for the average speed over the entire journey, both accelerating and braking sections. So this is what I did:

Final velocity after 400m:

x=(1/2)(u+v)t

400=(1/2)(0+v)19

v=42

It's not exactly 42 but I did use the exact answer for the rest of this. Anyway, I then did:

Braking distance:

x=(1/2)(42+0)5.1

x=107.1

So it should be 507.1/24.1=21 m/s

But when you do an acceleration/time graph and solve the area under the line you get the apparently correct answer of 26 m/s. I cannot for the life of me understand why or how my method doesn't work.