Find the displacement function if the acceleration function is a=2t+5. Assume a zero initial condition of displacement and v=8 when t=1.

Integrating the acceleration function gives the velocity function v, as below:
v = t2 +5t +C1, where C1 is a constant.

Integrating the velocity function gives the displacement function x, as below:
x = t3/3 + 5t2/2 + C1t + C2, where C2 is another constant.

The answer is completed by finding the 2 constants, C1 and C2.

With a zero initial condition of displacement, that means t=0, x=0. Put this initial condition into the displacement function ---> C2 = 0.

The boundary condition is that: v=8 when t=1. Simply put this condition into the velocity function ---> C1 = 2.

Thus, the complete displacement function is as below:
x =  t3/3 + 5t2/2 + 2t

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