Calculating Total Distance Traveled Based On Velocity-Time Graph

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To calculate the total distance traveled by a car moving along a straight path for 14 seconds, we need to analyze its velocity over time. This involves interpreting the provided velocity-time graph, which is crucial for understanding the car's motion and determining the distance covered. This comprehensive guide will walk you through the process step-by-step, ensuring a clear understanding of the concepts involved and the methods used to arrive at the final answer. Let's delve into the intricacies of distance calculation using velocity-time graphs, making this seemingly complex problem remarkably straightforward. Understanding the fundamental principles of kinematics is essential for solving this type of problem, and this guide aims to provide that understanding in a clear and concise manner.

Understanding Velocity-Time Graphs

The velocity-time graph is a powerful tool for visualizing the motion of an object. In this graph, the x-axis represents time (in seconds), and the y-axis represents velocity (in meters per second). The graph plots the car's velocity at different points in time, providing a comprehensive view of its motion throughout the 14-second journey. A positive velocity indicates movement in one direction, while a negative velocity indicates movement in the opposite direction. The magnitude of the velocity represents the speed of the car, with higher values indicating faster movement. The shape of the graph reveals important information about the car's motion; for instance, a straight line indicates constant acceleration, while a curved line indicates changing acceleration. The area under the velocity-time curve represents the displacement of the car, which is the change in its position. To find the total distance traveled, we need to consider both the positive and negative areas, as distance is a scalar quantity and does not depend on direction.

Analyzing the Provided Graph

Based on the provided graph, we can identify several key segments that represent different phases of the car's motion. The graph includes the following velocity data points:

  • At time t = 0 seconds, the velocity is 30 m/s.
  • At time t = 7 seconds, the velocity is 30 m/s.
  • From t = 0 to t = 1 second, the velocity decreases linearly from 30 m/s to -10 m/s.
  • At time t = 1 second, the velocity is -10 m/s.
  • From t = 1 to t = 5 seconds, the velocity remains constant at -10 m/s.
  • At time t = 5 seconds, the velocity is -10 m/s.
  • From t = 5 to t = 7 seconds, the velocity increases linearly from -10 m/s to 30 m/s.
  • From t = 7 to t = 14 seconds, the velocity is 30 m/s.

These data points allow us to break down the motion into distinct intervals, each with its own characteristics. Understanding the velocity profile during each interval is crucial for accurately calculating the total distance traveled. The linear changes in velocity indicate periods of constant acceleration or deceleration, while the constant velocity segments represent uniform motion. By examining these intervals separately, we can apply appropriate formulas to determine the distance traveled in each segment and then sum them up to find the total distance.

Calculating Distance in Different Intervals

To determine the total distance traveled, we need to calculate the distance covered in each distinct interval of the journey. This involves breaking down the 14-second period into segments where the car's motion is relatively uniform or changes predictably. Here's how we can approach the calculation for each segment:

  1. Interval 1: t = 0 to t = 1 second:
    • During this interval, the velocity changes linearly from 30 m/s to -10 m/s. This means the car is decelerating, coming to a stop, and then accelerating in the opposite direction.
    • To find the distance traveled, we can calculate the area under the velocity-time curve, which in this case is a trapezoid. However, since the velocity changes sign, we need to consider the areas above and below the x-axis separately.
    • First, find the time when the velocity is zero. Using the linear equation of the velocity change: v(t) = 30 - 40t. Setting v(t) = 0, we get t = 0.75 seconds.
    • The distance traveled while decelerating (0 to 0.75 seconds) is the area of the triangle above the x-axis: (0.5) * (0.75) * (30) = 11.25 meters.
    • The distance traveled while accelerating in the opposite direction (0.75 to 1 second) is the area of the triangle below the x-axis: (0.5) * (0.25) * (10) = 1.25 meters.
    • The total distance traveled in this interval is 11.25 + 1.25 = 12.5 meters.
  2. Interval 2: t = 1 to t = 5 seconds:
    • In this interval, the velocity remains constant at -10 m/s, indicating uniform motion in the opposite direction.
    • The distance traveled is simply the product of the speed and the time: |-10 m/s| * (5 - 1) s = 10 * 4 = 40 meters.
  3. Interval 3: t = 5 to t = 7 seconds:
    • Here, the velocity increases linearly from -10 m/s to 30 m/s. This means the car is accelerating back towards its original direction.
    • The distance traveled is the area of the trapezoid formed by the velocity-time graph. We can also think of this as the sum of the areas of a rectangle and a triangle.
    • The distance traveled can be calculated using the average velocity method: ((30 + 10) / 2) * (7 - 5) = 20 * 2 = 40 meters.
  4. Interval 4: t = 7 to t = 14 seconds:
    • During this interval, the velocity is constant at 30 m/s, indicating uniform motion in the original direction.
    • The distance traveled is the product of the speed and the time: 30 m/s * (14 - 7) s = 30 * 7 = 210 meters.

By calculating the distance for each segment, we have a clear picture of how the car moved throughout its journey. Each interval presents a unique aspect of the motion, whether it's uniform motion, acceleration, or deceleration. Understanding these individual components is essential for the final calculation of the total distance traveled.

Calculating the Total Distance Traveled

Now that we have calculated the distance traveled in each interval, we can determine the total distance traveled by summing up the distances from each segment. This is a straightforward process that combines the results of our previous calculations to provide the final answer. Remember, distance is a scalar quantity, so we add the magnitudes of the distances traveled in each segment, regardless of direction.

  • Distance in Interval 1: 12.5 meters
  • Distance in Interval 2: 40 meters
  • Distance in Interval 3: 40 meters
  • Distance in Interval 4: 210 meters

Adding these distances together, we get:

Total Distance = 12.5 m + 40 m + 40 m + 210 m = 302.5 meters

Therefore, the total distance traveled by the car over the 14 seconds is 302.5 meters. This calculation represents the sum of all the movements the car made, regardless of direction. This final result provides a comprehensive measure of the car's journey, taking into account all its changes in velocity and direction.

Conclusion

In conclusion, determining the total distance traveled by a car involves a thorough analysis of its velocity-time graph. By breaking down the motion into distinct intervals, calculating the distance traveled in each segment, and then summing these distances, we arrive at the total distance. In this case, the car traveled a total of 302.5 meters in 14 seconds. This process highlights the importance of understanding velocity-time graphs and their relationship to motion. The ability to interpret these graphs and perform these calculations is a fundamental skill in physics and engineering. This guide has provided a detailed explanation of the method, ensuring that readers can confidently apply these principles to similar problems in the future. By mastering these techniques, you can gain a deeper understanding of kinematics and the motion of objects in various scenarios.

Therefore, the total distance traveled by the car is 302.5 meters.