To determine the time at which the horse was traveling the fastest, we need to analyze the changes in position over time and calculate the velocities between each consecutive pair of time intervals. Below is a detailed step-by-step solution:
1. Identify the time intervals and their corresponding positions:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Time (s)} & \text{Position (m)} \\
\hline
0 & 5 \\
\hline
1 & 9 \\
\hline
2 & 13 \\
\hline
3 & 13 \\
\hline
4 & 15 \\
\hline
5 & 20 \\
\hline
\end{array}
\][/tex]
2. Calculate the differences in position (distance) between each consecutive pair of time intervals to find the velocities:
[tex]\[
\begin{align*}
\text{Velocity from 0 to 1 s} &= \frac{9 - 5}{1 - 0} = 4 \, \text{m/s} \\
\text{Velocity from 1 to 2 s} &= \frac{13 - 9}{2 - 1} = 4 \, \text{m/s} \\
\text{Velocity from 2 to 3 s} &= \frac{13 - 13}{3 - 2} = 0 \, \text{m/s} \\
\text{Velocity from 3 to 4 s} &= \frac{15 - 13}{4 - 3} = 2 \, \text{m/s} \\
\text{Velocity from 4 to 5 s} &= \frac{20 - 15}{5 - 4} = 5 \, \text{m/s} \\
\end{align*}
\][/tex]
3. List the calculated velocities:
[tex]\[
[4.0, 4.0, 0.0, 2.0, 5.0] \, \text{m/s}
\][/tex]
4. Identify the maximum velocity from the list:
[tex]\[
\text{Max velocity} = 5.0 \, \text{m/s}
\][/tex]
5. Determine the time interval corresponding to this maximum velocity:
The maximum velocity of [tex]\(5.0 \, \text{m/s}\)[/tex] occurs between 4 and 5 seconds. Therefore, the horse was traveling the fastest at 5 seconds.
Given the options presented (1 s, 2 s, 3 s, 4 s), none of them are correct. However, if the correct time intervals would be available, the horse was traveling the fastest at 5 seconds.
If we have to choose among the given options, we could say that there is no enough information for a correct answer.
