Nick starts jogging and runs faster and faster for 3...

Nick starts jogging and runs faster and faster for 3 minutes, then he walks for 5 minutes. He stops at an intersection for 2 minutes, runs fairly quickly for 5 minutes, then walks for 4 minutes. $$\begin{array}{l}{\text { (a) Sketch a possible graph of the distance } s \text { Nick has covered }} \\ {\text t \text { minutes. }} \\ {\text { (b) Sketch a graph of } d s / d t}\end{array}$$

Nick starts jogging and runs faster and faster for 3 minutes, then he walks for 5 minutes. He stops at an intersection for 2 minutes, runs fairly quickly for 5 minutes, then walks for 4 minutes.
$$\begin{array}{l}{\text { (a) Sketch a possible graph of the distance } s \text { Nick has covered }} \\ {\text t \text { minutes. }} \\ {\text { (b) Sketch a graph of } d s / d t}\end{array}$$

Key Concepts

Graphical Interpretation of Motion

Understanding the connection between the physical motion and its graphical representations is crucial. This involves linking specific segments of a graph (like rising segments for increasing speed, flat segments for periods of rest, or step changes) to the corresponding motion behaviors. Graphical interpretation allows one to visualize and analyze complex real-world motions using key concepts from calculus and kinematics.

Distance-Time Graph

A distance-time graph represents how an object’s position or the distance it has traveled changes over time. In motion problems, changes in the slope of this graph indicate different phases of movement, such as acceleration (increasing slope), deceleration or slowing down (decreasing slope), constant speed (a straight line with constant slope), or rest (horizontal line). This graph serves as a visual tool to understand and analyze various stages of motion.

Derivative as Velocity

The derivative of the distance function with respect to time gives the velocity function. This means that the slope of the distance-time graph at any point corresponds to the instantaneous speed of the moving object. A positive slope indicates forward motion, a zero slope shows that there is no movement, and a negative slope would indicate reversing direction. Graphing the derivative provides insights into how the speed changes over time.

Piecewise Functions

Motion problems often involve different intervals with distinct behaviors such as acceleration, constant speed, deceleration, or stops. Modeling these intervals as piecewise functions helps to accurately capture the behavior of the object in each segment. Each segment of the function can be represented by a different rule or formula, making it easier to analyze and graph the overall motion.

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