Completing Gracie's 200m Race Distance-Time Graph A Comprehensive Guide

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Gracie's 200m race is an excellent example to illustrate the relationship between distance, time, and speed in a graphical format. Analyzing her performance through a distance-time graph allows us to understand her pace and consistency throughout the race. The first part of Gracie's race is already depicted in a distance-time graph, and we know she completed the remaining distance at a constant speed of 5m/s. Our task is to determine which line, among options A to D, correctly completes Gracie's distance-time graph.

Understanding Distance-Time Graphs

Before we dive into the specifics of Gracie's race, let's briefly review the key concepts of distance-time graphs. A distance-time graph plots the distance traveled by an object against time. The y-axis represents the distance, usually measured in meters (m), while the x-axis represents the time, typically measured in seconds (s). The slope of the line at any point on the graph represents the speed of the object at that instant. A steeper slope indicates a higher speed, while a flatter slope indicates a lower speed. A horizontal line signifies that the object is stationary.

In Gracie's case, the initial part of the graph shows her varying speed during the first portion of the race. The remaining part, which we need to determine, represents her constant speed of 5m/s. This constant speed will be depicted as a straight line with a consistent slope. To find the correct line, we need to calculate the time it took Gracie to cover the remaining distance at 5m/s and then plot it on the graph.

Analyzing the Initial Graph Section

The provided distance-time graph for the first part of Gracie's race is crucial for determining the starting point for the remaining part of her race. We need to carefully analyze this section to extract the following information:

  • Distance covered: Determine the distance Gracie covered in the first part of the race. This can be read directly from the graph where the initial segment ends on the y-axis (distance).
  • Time taken: Determine the time Gracie took to cover that initial distance. This can be read from the graph where the initial segment ends on the x-axis (time).

Let's assume, for example, that the graph shows Gracie covered 120 meters in the first 20 seconds. This means she has 80 meters left to run (200m total - 120m covered). This also gives us a starting point of (20 seconds, 120 meters) on the graph for the remaining part of her race. This starting point is crucial as we need to ensure the line we choose continues from this point seamlessly.

Calculating Time for the Remaining Distance

Now that we know the remaining distance and Gracie's constant speed, we can calculate the time it took her to cover the rest of the race. We can use the following formula:

Time = Distance / Speed

Using our example, where the remaining distance is 80 meters and the speed is 5 m/s:

Time = 80 meters / 5 m/s = 16 seconds

This means it took Gracie 16 seconds to run the remaining 80 meters. Now we know that the line we choose must cover the remaining distance in 16 seconds.

Identifying the Correct Line

To identify the correct line (A, B, C, or D), we need to consider the following:

  1. Starting Point: The line must start from the point where the initial graph section ends (e.g., 20 seconds, 120 meters).
  2. Slope: The line must have a constant slope representing a speed of 5 m/s. This means for every 1 second increase in time, the distance should increase by 5 meters.
  3. Ending Point: The line must reach the 200-meter mark on the distance axis. We calculated that it takes 16 seconds to cover the remaining distance, so the line should end 16 seconds after the time at which the first part of the graph ended (e.g., if the first part ended at 20 seconds, the line should end at 36 seconds).

By visually inspecting the lines A to D, we can eliminate those that do not meet these criteria. For example, a line that starts at the wrong point, has a different slope (indicating a speed other than 5 m/s), or does not reach the 200-meter mark within the calculated time frame would be incorrect.

Determining the Correct Line: A Step-by-Step Approach

Let's break down the process of identifying the correct line into a series of steps:

  1. Read the graph: Carefully examine the initial part of the graph to determine the distance covered and time taken during the first segment of Gracie's race. Note the coordinates (time, distance) where this segment ends.
  2. Calculate remaining distance: Subtract the distance covered in the first part from the total race distance (200 meters) to find the remaining distance.
  3. Calculate time for remaining distance: Use the formula Time = Distance / Speed with the remaining distance and Gracie's constant speed of 5 m/s to calculate the time taken for the second segment.
  4. Determine the endpoint: Add the time calculated in step 3 to the time at which the first segment ended to find the total time taken for the race. The endpoint of the correct line should correspond to (total time, 200 meters).
  5. Evaluate the slope: The correct line should have a constant slope representing a speed of 5 m/s. To check this, pick two points on the line and calculate the slope using the formula Slope = (Change in Distance) / (Change in Time). The slope should be approximately 5.
  6. Compare with options: Compare the starting point, slope, and ending point of each line (A to D) with the values calculated in the previous steps. The line that matches all criteria is the correct line.

Potential Challenges and Solutions

  • Reading the graph accurately: Distance-time graphs can sometimes be tricky to read accurately. Use a ruler or straight edge to help align points and read values from the axes precisely. Double-check your readings to avoid errors.
  • Calculating the slope: Ensure you choose two distinct points on the line to calculate the slope accurately. Avoid choosing points that are too close together, as this can lead to inaccuracies.
  • Visual estimation: Sometimes, distinguishing between lines with slightly different slopes can be challenging visually. Focus on the calculated time and distance values to confirm the correctness of the line.

Conclusion

Completing Gracie's distance-time graph requires a thorough understanding of distance-time graph principles and careful calculations. By accurately analyzing the initial graph segment, calculating the time for the remaining distance, and evaluating the slope, we can confidently identify the correct line that completes Gracie's race profile. Understanding these concepts is crucial not only for solving this specific problem but also for interpreting and analyzing motion in various real-world scenarios. Remember to break down the problem into smaller, manageable steps, and always double-check your calculations and readings to ensure accuracy. The ability to interpret and create distance-time graphs is a valuable skill in mathematics and physics, providing a visual representation of motion and speed over time.