A particle is moving with the given data. Find the position of the particle. A particle is moving with the given data. Find the position of the particle. a(t) = t^2 − 9t + 6,   s(0) = 0,   s(1) = 20   s(t)= ___________

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You need to remember that you need to integrate the velocity of particle to find the position function.

Since the problem provides the acceleration function,you need to integrate this function to find velocity function and then you need to integrate velocity to find position function such that:

`v(t) = int a(t) dt`

You need to substitute `t^2 - 9t + 6`  for `a(t) `  such that:

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

You need to split the integral into simpler integrals such that:

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

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

You need to integrate v(t) to find position function such that:

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

You need to split the integral into simpler integrals such that:

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

`s(t) = t^4/12 - 3t^2/2 + 3t^2 + ct + d`

Notice that the position function is not determined since it contains two unknown coefficients c and d.

The problem provides two informations such that:

`s(0) = 0 =gt d = 0`

`s(1) = 20 =gt 1/12 - 3/2 + 3 + c = 20`

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

`c = (240 - 1 + 18 - 36)/12`

`c = 221/12`

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

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