91Ó°ÊÓ

Velocity gives us an idea of how quickly something moves in a certain direction. In simpler terms, it's the rate at which an object covers distance. In the context of our marathon exercise, the runner's velocity changes for each segment of the race. Understanding these changes is key to building a position-time graph.

To calculate velocity in our example, the runner maintains an average velocity of:

• 15.0 km/h for the first 10 km
• 10.0 km/h for the next 15 km
• 5.0 km/h for the last 15 km

When graphed, the velocity for each segment is represented by the slope of the line in a position-time graph. A steeper slope indicates a higher velocity, while a more gentle slope shows a slower velocity. Each segment of a marathon is a straight line due to constant speeds, which makes it easier to interpret velocity changes.

As the runner progresses through different segments of the marathon, it's important to keep track of time. This is where cumulative time comes into play. Cumulative time is simply the total time elapsed up to a certain point in the race.

For each segment, we've already calculated time separately:

• 0.67 hours for the first 10 km.
• 1.5 hours for the middle 15 km.
• 3.0 hours for the last 15 km.

Now, to find cumulative time:

• After 10 km: Start with 0.67 hours.
• After 25 km: Add 0.67 hours and 1.5 hours to get 2.17 hours.
• After 40 km: Add 2.17 hours and 3.0 hours to reach a total of 5.17 hours.

This cumulative tracking is important for plotting points accurately on the position-time graph and understanding the overall race duration.

Distance calculation forms the crux of our analysis when constructing a position-time graph. By knowing how far the runner travels in each segment, combined with the cumulative time, we can successfully depict the runner's progress.

Here's how we consider the distance:

• The first segment covers 10 km.
• The second segment increases the distance to 25 km.
• The final segment brings the runner's journey to 40 km.

These distances correlate with specific points on the graph, with time on the x-axis and distance on the y-axis. With each segment maintaining a linear pattern, it's clear how the graph illustrates the runner's constant velocity sections. Each plotted point—whether (0.67, 10 km), (2.17, 25 km), or (5.17, 40 km)—shows both time and cumulative distance clearly, aiding in understanding the motion.