Write an expression for the particles velocity and an expression for the position s(t) of the particle

Question

A particle moves along x axis so that it’s acceleration any time t>0 is given by a(t)=12t-18. At time t=1 the velocity of the particle is v(1)=0 and s(1)=9

Answer 1

a = dv/dt

a dt = dv

∫a dt = ∫dv

∫12t-18 dt = ∫dv

6t² - 18t + C = v(t)

But we are given that v(1) = 0 so

6(1)² - 18(1) + C₁ = 0

C₁ = 18 - 6 = 12

So v(t) = 6t² - 18t + 12

v = ds/dt

v dt = ds

∫v dt = ∫ds

∫6t² - 18t + 12 dt = ∫ds

2t³ - 9t² + 12t + C₂ = s(t)

But we are given that s(1)=9 so

2(1)³ - 9(1)² + 12(1) + C₂ = 9

C₂ = 9 - 2 + 9 - 12 = 4

So s(t) = 2t³ - 9t² + 12t + 4