To find the position function [tex]\( s(t) \)[/tex] for an object moving along a line with a given velocity function [tex]\( v(t) = 8t + 3 \)[/tex] and an 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 found by integrating the velocity function [tex]\( v(t) \)[/tex] with respect to time [tex]\( t \)[/tex].
[tex]\[
s(t) = \int v(t) \, dt
\][/tex]
So we have:
[tex]\[
s(t) = \int (8t + 3) \, dt
\][/tex]
2. Compute the Integral:
[tex]\[
\int 8t \, dt = 8 \cdot \frac{t^2}{2} = 4t^2
\][/tex]
[tex]\[
\int 3 \, dt = 3t
\][/tex]
Adding these results, we get:
[tex]\[
s(t) = 4t^2 + 3t + C
\][/tex]
where [tex]\( C \)[/tex] is the constant of integration.
3. Determine the Constant of Integration:
Given that the initial position [tex]\( s(0) = 0 \)[/tex], we can find the value of [tex]\( C \)[/tex].
[tex]\[
s(0) = 4(0)^2 + 3(0) + C = 0
\][/tex]
From this, we see that:
[tex]\[
C = 0
\][/tex]
4. Write the Final Position Function:
Therefore, the position function [tex]\( s(t) \)[/tex] is:
[tex]\[
s(t) = 4t^2 + 3t
\][/tex]
So, the position function is given by:
[tex]\[
s(t) = 4t^2 + 3t
\][/tex]
