If we know the acceleration function for the moving object i

If we know the acceleration function for the moving object is a(t) = 12t + 1, and we also know that at the lime t = 0, the object moving at the velocity of 1 mph from the ground, please find the velocity function and the distance function.

Solution

a(t) = 12t+1
v(t) = integral(12t+1)
= 6t²+t+c

v(0) = 1 so c = 1

v(t) = 6t²+t+1

s(t) = integral(6t²+t+1)

      = 2t³+t²/2+t+c

at t=0 a=1,v=1

s(0)=v²/2a = 1/2 so c=1/2

s(t)=2t³+t²/2+t+1/2