# A particle moves on a straight line with velocity function v(t)= sin((alpha)t)cos^2((alpha)t). Find its position function s(t) if s(0) = 2.

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember the equation that relates the velocity function and position function such that:

`v(t) = s'(t) => int v(t) dt = s(t) + c`

Hence, you need to integrate the given velocity function to find the position function `s(t)`  such that:

`int sin^2(alpha*t)cos^2(alpha*t) dt = s(t) + c`

You should use the half angle identities such that:

`sin^2(alpha*t) = (1 - cos 2alpha*t)/2`

`cos^2(alpha*t) = (1+ cos 2alpha*t)/2`

`int sin^2(alpha*t)cos^2(alpha*t) dt =(1/4) int (1 - cos 2alpha*t)(1+ cos 2alpha*t) dt`

Converting the product into a difference of squares yields:

`int sin^2(alpha*t)cos^2(alpha*t) dt = (1/4) int (1 - cos^2 2alpha*t) dt`

Using linearity yields:

`int sin^2(alpha*t)cos^2(alpha*t) dt = (1/4) int dt - (1/4) int cos^2 2alpha*t dt`

Using the half angle identity yields:

`cos^2 2alpha*t = (1 + cos 4alpha*t)/2`

`int sin^2(alpha*t)cos^2(alpha*t) dt = (1/4) int dt - (1/8)(int dt + int cos 4alpha*t dt)`

`int sin^2(alpha*t)cos^2(alpha*t) dt = (1/4)t - (1/8)t - (1/8)(sin(4alpha*t))/(4 alpha) + c`

`int sin^2(alpha*t)cos^2(alpha*t) dt = t/8 - (sin(4alpha*t))/(32 alpha) + c`

Using the information provided by the problem that `s(0) = 2` , you may find the constant c such that:

`s(0) = 0 - (sin 0)/(32 alpha) + c`

`2 = c`

Hence, evaluating the position function under the given conditions yields `s(t) = t/8 - (sin(4alpha*t))/(32 alpha) + 2.`