Given the following velocity function of an object moving along a line, find the position function with the given initial position. \( v(t)=6 t+4 ; s(0)=0 \) The position function is \( s(t)=\square \)

To find the position function, we simply need to integrate the velocity function \( v(t) = 6t + 4 \). When we integrate, we get \( s(t) = \int (6t + 4) dt = 3t^2 + 4t + C \). To determine the constant \( C \), we use the initial condition \( s(0) = 0 \). Plugging in \( t = 0 \) gives \( 0 = 3(0)^2 + 4(0) + C \), therefore, \( C = 0 \). Thus, the position function is \( s(t) = 3t^2 + 4t \). Now you've got that position function, but did you know that integrating functions is a bit like piecing together a puzzle? Each part you add helps you see the bigger picture of motion over time! And talking about motion, did you realize that velocity functions like \( v(t) = 6t + 4 \) can model real-world scenarios? For example, a car accelerating over time would have a changing velocity similar to this equation, allowing us to predict how far it travels over a given time – perfect for planning those road trips!