Question

A particle is moving with the given data. Find the position of the particle.
a(t) = t^2 − 8t + 5, s(0) = 0, s(1) = 20

Answer 1

are you using the usual notation
so that a(t) is acceleration, v(t) is velocity and s(t) is distance ??

I will assume you are

a(t) = t^2 - 8t + 5
v(t) = (1/3)t^3 - 4t^2 + 5t + c
s(t) = (1/12)t^4 - (4/3)t^3 + (5/2)t^2 + ct + k

s(0) = 0
0-0+0+0+k=0
k=0
s(1) = 20
1/12 - 4/3 + 5/2 + c = 0 = 20
5/4 + c = 20
c = 75/4

s(t) = (1/12)t^4 - (4/3)t^3 + (5/2)t^2 + (75/4)t