Find $s(t)$, where $s(t)$ represents the position function, $v(t)$ represents the velocity function, and $\mathrm{a}(\mathrm{t})$ represents the acceleration function.
\[
a(t)=-24 t+8 \text {, with } v(0)=1 \text { and } s(0)=9
\]
Answer
Thus, the position function \(s(t) = -4t^3 + 4t^2 + t + 9\) and the velocity function \(v(t) = -12t^2 + 8t + 1\) satisfy the given conditions, and the solution is \(\boxed{s(t) = -4t^3 + 4t^2 + t + 9}\).
Step 1 ：Given the acceleration function \(a(t) = -24t + 8\), we can find the velocity function \(v(t)\) by integrating \(a(t)\) with respect to \(t\).
Step 2 ：Performing the integration, we get \(v(t) = -12t^2 + 8t + C\), where \(C\) is the constant of integration.
Step 3 ：We are given that \(v(0) = 1\), so we can substitute \(t = 0\) into the equation to solve for \(C\).
Step 4 ：Substituting \(t = 0\) into the equation, we get \(1 = -12(0)^2 + 8(0) + C\), which simplifies to \(C = 1\).
Step 5 ：So, the velocity function is \(v(t) = -12t^2 + 8t + 1\).
Step 6 ：Next, we can find the position function \(s(t)\) by integrating \(v(t)\) with respect to \(t\).
Step 7 ：Performing the integration, we get \(s(t) = -4t^3 + 4t^2 + t + D\), where \(D\) is the constant of integration.
Step 8 ：We are given that \(s(0) = 9\), so we can substitute \(t = 0\) into the equation to solve for \(D\).
Step 9 ：Substituting \(t = 0\) into the equation, we get \(9 = -4(0)^3 + 4(0)^2 + (0) + D\), which simplifies to \(D = 9\).
Step 10 ：So, the position function is \(s(t) = -4t^3 + 4t^2 + t + 9\).
Step 11 ：Finally, we check our results by substituting \(t = 0\) into the position function and the velocity function. We find that \(s(0) = 9\) and \(v(0) = 1\), which are the given initial conditions.
Step 12 ：Thus, the position function \(s(t) = -4t^3 + 4t^2 + t + 9\) and the velocity function \(v(t) = -12t^2 + 8t + 1\) satisfy the given conditions, and the solution is \(\boxed{s(t) = -4t^3 + 4t^2 + t + 9}\).
