# Let s(t) represent the position of a particle at time t, v(t) is the velocity and a(t) is the acceleration. What is the value of v(9) given that a(t) = -3t^2+ 3t + 10, s(1) = 1 and s(2) = 1.

justaguide | College Teacher | (Level 2) Distinguished Educator

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The position of the particle is given by s(t), the velocity by v(t) and the acceleration by a(t).

It should be kept in mind that v(t) is the derivative of displacement, not of position.

As a function of acceleration `v(t) = int a(t) dt`

=> `int -3*t^2 + 3t + 10 dt`

=> `-3*t^3/3 + 3*t^2/2 + 10t + C_1`

Displacement d(t) as a function of v(t) is `d(t) = int v(t) dt`

=> `int -t^3 + (3/2)*t^2 + 10t + C_1 dt`

=> `-t^4/4 + t^3/2 + 5t^2 + C_1*t + C_2`

The position at s(1) = 1 and s(2) = 1.

The displacement from t = 1 to t = 2 is 0

=> `-4 + 4 + 20 + 2*C_1 + C_2 - (-1/4 + 1/2 + 5 + C_1 + C_2) = 0`

=> `20 + 2C_1 + C_2 - 5.25 - C_1 - C_2 = 0`

=> `14.75 + C_1 = 0`

=> `C_1 = -14.75`

It is not possible to find `C_2` but as the value of v(9) is required that does not pose a constraint.

`v(9) = -9^3 + 3*9^2/2 + 10*9 - 14.75 = -532.25`

The value of v(9) = -532.25