Find $s(t)$, where $s(t)$ represents the position function, $v(t)$ represents the velocity function, and a(t) represents the acceleration function. \[ a(t)=-6 t+2 \text {, with } v(0)=2 \text { and } s(0)=9 \] \[ s(t)= \]
Solution
Step 1 ：Given the acceleration function \(a(t) = -6t + 2\).
Step 2 ：The velocity function \(v(t)\) is the integral of the acceleration function. Integrate \(a(t)\) to get \(v(t) = -3t^2 + 2t + C_1\).
Step 3 ：Given the initial condition \(v(0) = 2\), substitute into the velocity function to solve for \(C_1\). This gives \(C_1 = 2\).
Step 4 ：So the velocity function is \(v(t) = -3t^2 + 2t + 2\).
Step 5 ：The position function \(s(t)\) is the integral of the velocity function. Integrate \(v(t)\) to get \(s(t) = -t^3 + t^2 + 2t + C_2\).
Step 6 ：Given the initial condition \(s(0) = 9\), substitute into the position function to solve for \(C_2\). This gives \(C_2 = 9\).
Step 7 ：So the position function is \(s(t) = -t^3 + t^2 + 2t + 9\).
Step 8 ：\(\boxed{s(t) = -t^3 + t^2 + 2t + 9}\) is the final answer.
