To find the position function [tex]\( s(t) \)[/tex] from the given velocity function [tex]\( v(t) \)[/tex] and initial position [tex]\( s(0) = 0 \)[/tex], we need to follow these steps:
1. Integrate the velocity function: The position function [tex]\( s(t) \)[/tex] is the integral of the velocity function [tex]\( v(t) \)[/tex]. Given:
[tex]\[
v(t) = 6t^2 + 6t - 9
\][/tex]
we will integrate [tex]\( v(t) \)[/tex] with respect to time [tex]\( t \)[/tex].
2. Compute the indefinite integral: Perform the integration of [tex]\( v(t) \)[/tex] term by term.
[tex]\[
\int (6t^2 + 6t - 9) \, dt
\][/tex]
- Integrate [tex]\( 6t^2 \)[/tex]: [tex]\(\int 6t^2 \, dt = 2t^3 \)[/tex]
- Integrate [tex]\( 6t \)[/tex]: [tex]\(\int 6t \, dt = 3t^2 \)[/tex]
- Integrate [tex]\(-9 \)[/tex]: [tex]\(\int -9 \, dt = -9t \)[/tex]
Combining these results, we get:
[tex]\[
\int (6t^2 + 6t - 9) \, dt = 2t^3 + 3t^2 - 9t + C
\][/tex]
where [tex]\( C \)[/tex] is the constant of integration.
3. Apply the initial condition: Use the initial position [tex]\( s(0) = 0 \)[/tex] to determine the constant [tex]\( C \)[/tex].
[tex]\[
s(0) = 2(0)^3 + 3(0)^2 - 9(0) + C = 0
\][/tex]
This simplifies to [tex]\( C = 0 \)[/tex].
4. Write the final position function: Substitute [tex]\( C = 0 \)[/tex] back into the expression for [tex]\( s(t) \)[/tex].
[tex]\[
s(t) = 2t^3 + 3t^2 - 9t
\][/tex]
Therefore, the position function [tex]\( s(t) \)[/tex] is:
[tex]\[
s(t) = 2t^3 + 3t^2 - 9t
\][/tex]
