# Find the position of the particle.  a(t) = t2 − 9t + 6,    s(0) = 0,    s(1) = 20   s(t)= A particle is moving with the given data. Find the position of the particle.   a(t) = t2 − 9t + 6,    s(0) = 0,    s(1) = 20   s(t)=_______________

You need to remember that position of particle may be found if the speed is given such that:

`s = v*t`

The problem provides you the function of acceleration, hence you should remember that `a(t) = (dv)/(dt),`  hence `a(t)dt = dv` . You need to integrate the equation `a(t)dt = dv`  such that:

`int a(t) dt = int dv =gt int (t^2-9t+6)dt=v(t)`

`v(t) = int t^2 dt - int 9t dt + int 6dt`

`v(t) = t^3/3 - 9t^2/2 + 6t + c_1`

You need to use the relation  `v(t) = (ds)/(dt)`  such that:

`ds = v(t)dt =gt int ds = int v(t)` dt

`s(t) = int(t^3/3 - 9t^2/2 + 6t + c_1) dt`

`s(t) = t^4/12 - 9t^3/6 + 6t^2/2 + c_1*t + c_2`

`s(t) = t^4/12 - 3t^3/2 + 3t^2 + c_1*t + c_2`

The problem provides the informations s(0)=0 and s(1)=20, hence, substituting 0 and 1 for t in s(t) yields:

`c_2=0`

`1/12 - 3/2 + 3 + c_1 = 20`

`c_1 = 17 + 3/2 - 1/12`

`c_1 = (204 + 18 - 1)/12`

`c_1 = 221/12`

Hence, evaluating the function s(t) under given conditions yields `s(t) = t^4/12 - 3t^3/2 + 3t^2 + 221t/12.`

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