Given the following velocity function of an object moving along a line, find the position function with the given initial position.

[tex]\[ v(t) = 8t + 3 \][/tex]
[tex]\[ s(0) = 0 \][/tex]

The position function is [tex]\( s(t) = \boxed{\phantom{}} \)[/tex].



Answer :

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]