Question

three curves that describe the motion of a particle along a coordinate axis at time t (in seconds) are shown below. they are polynomial functions related to each other through differentiation of the position function s(t) (which has units of meters). each of your selected curves should have the color that corresponds to your chosen function: orange for s(t), blue for v(t), and green for a(t). provide your answer below.

Explanation:

Step1: Recall the relationships

The velocity function $v(t)$ is the derivative of the position - function $s(t)$, i.e., $v(t)=s^{\prime}(t)$. The acceleration function $a(t)$ is the derivative of the velocity function $v(t)$, i.e., $a(t)=v^{\prime}(t)=s^{\prime\prime}(t)$.

Step2: Analyze the slopes

To find $v(t)$ from $s(t)$, we look at the slope of $s(t)$ at different points. To find $a(t)$ from $v(t)$, we look at the slope of $v(t)$ at different points. We need to estimate the slopes of the curves at various $t$ - values to match the functions.

Step3: Determine the functions

By observing the slopes of the position - function $s(t)$ (orange curve) to get the velocity function $v(t)$ (blue curve), and then observing the slopes of the velocity function $v(t)$ to get the acceleration function $a(t)$ (green curve). We need to provide the actual functions based on the shape of the curves. If the position function $s(t)$ is a polynomial of degree $n$, then $v(t)$ is a polynomial of degree $n - 1$ and $a(t)$ is a polynomial of degree $n - 2$. For example, if $s(t)=at^{3}+bt^{2}+ct + d$, then $v(t)=3at^{2}+2bt + c$ and $a(t)=6at+2b$. Without the actual data points on the curves, we can only describe the general process of finding the functions. We would need to use techniques like curve - fitting (if the exact form of the polynomial is required) or just qualitative analysis of slopes for a general description.

Since no data points are given to determine exact polynomial functions, we assume we are doing a qualitative analysis. If the position function $s(t)$ has an increasing slope in some interval, then $v(t)>0$ in that interval. If the slope of $s(t)$ is increasing, then $a(t)>0$ in that interval.

Answer:

Without specific data points on the curves, we cannot provide exact polynomial functions for $s(t)$, $v(t)$, and $a(t)$. But the general relationships are: $v(t)$ is the first - derivative of $s(t)$ and $a(t)$ is the first - derivative of $v(t)$ (or the second - derivative of $s(t)$). To find the functions, analyze the slopes of the given curves at different points.