Shown are graphs of the position functions of two runners, A...

Shown are graphs of the position functions of two runners,
A and B, who run a 100 -meter race and finish in a tie.
(a) Describe and compare how the runners run the race.
(b) At what time is the distance between the runners the
greatest?
(c) At what time do they have the same velocity?

Key Concepts

Position Functions

A position function represents the location of an object as it changes over time. In the context of motion, it allows one to track and compare the progress of objects or individuals along a path, such as runners in a race. Understanding position functions is crucial for analyzing where each runner is at any given moment during the race.

Velocity Functions

Velocity functions are derived from position functions and represent the rate of change of position with respect to time. In these types of problems, comparing the velocity functions of two moving objects helps to determine their speeds, analyze when they might be accelerating or decelerating, and identify moments when they have the same instantaneous speed.

Relative Distance Analysis

Relative distance analysis involves examining the difference between the position functions of two objects. By comparing positions directly, one can determine at which point in time the separation between the runners is greatest. This concept is important for problems requiring the identification of maximum or minimum differences in distance between moving objects.

Graphical Interpretation of Motion

Graphical interpretation involves using the visual representations of the position functions to understand and compare different aspects of motion, such as speed, acceleration, and overall performance. Graph analysis serves as a tool to interpret when runners might be leading or trailing, where the greatest separation occurs, and when they may be moving at equivalent speeds.

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