Problem

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.

From Solvely APP
Source: https://solvelyapp.com/problems/7942/

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