Understanding Speed from a Distance-Time Graph

A distance-time graph is a powerful tool used to represent the motion of an object. In such a graph, the distance traveled by the object is plotted on the vertical axis (y-axis), and time taken is plotted on the horizontal axis (x-axis).

The speed of an object moving at a constant velocity is given by the change in distance divided by the change in time. On a distance-time graph, the speed is represented by the slope of the line segment during a particular interval.

The formula for slope ($m$) of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

\(m = \frac{y_2 - y_1}{x_2 - x_1}\)

In a distance-time graph, if distance is on the y-axis and time on the x-axis, the slope is:

\(\text{Slope} = \frac{\text{Change in Distance}}{\text{Change in Time}}\)

This is exactly the definition of speed. Therefore, the slope of the line segment in a distance-time graph represents the speed of the object during that time interval.

• A steeper slope indicates a higher speed.
• A less steep slope indicates a lower speed.
• A horizontal line indicates zero speed (the object is stationary).

Let's analyze the given graph which represents a car journey in three parts: A, B, and C. We need to compare the speeds \(V_a\), \(V_b\), and \(V_c\) in these respective parts.

Analyzing Each Part of the Car Journey

We will visually inspect the slope of the line segments in parts A, B, and C to determine the relative speeds.

• Part A: The line segment A starts from the origin and moves upwards and to the right. It has a positive slope, indicating the car is moving away from the starting point at a constant speed \(V_a\).

• Part B: The line segment B starts where A ends and continues upwards and to the right. Compared to segment A, segment B is noticeably steeper. This indicates a higher slope than A, meaning the car is moving at a higher speed \(V_b\) in part B.

• Part C: The line segment C starts where B ends and continues upwards and to the right. Compared to segment B, segment C is much less steep. Compared to segment A, segment C is also less steep. This indicates a lower slope than both A and B, meaning the car is moving at a lower speed \(V_c\) in part C.

Comparing the Speeds $V_a$, $V_b$, and $V_c$

Based on our analysis of the slopes:

• The slope of B is the steepest.
• The slope of A is less steep than B but steeper than C.
• The slope of C is the least steep.

Since speed is represented by the slope in a distance-time graph, we can conclude:

\(\text{Slope of B} > \text{Slope of A} > \text{Slope of C}\)

Therefore, the speeds are related as:

\(V_b > V_a > V_c\)

Conclusion

Comparing this relationship with the given options, we find that the correct relation between the speeds in parts A, B, and C of the car journey is \(V_b > V_a > V_c\).

Revision Table: Distance-Time Graph Interpretation

Additional Information: Types of Motion Graphs

Besides distance-time graphs, velocity-time graphs and acceleration-time graphs are also commonly used to describe motion. Each type of graph provides different information about the motion of an object.

• Velocity-Time Graphs:

  • The slope represents acceleration.
  • The area under the graph represents displacement.

• Acceleration-Time Graphs:

  • The area under the graph represents the change in velocity.

Understanding how to interpret the slope and area for each type of graph is crucial for analyzing motion.