Problem 53 - Brief Applied Calculus

Question

Velocity Each of the following three "stories," labeled \(\mathbf{a}, \mathbf{b}\), and \(\mathbf{c}\), matches one of the velocity graphs, labeled (i), (ii), and (iii). For each story, choose the most appropriate graph. a. I left my home and drove to meet a friend, but I got stopped for a speeding ticket. Afterward I drove on more slowly. b. I started driving but then stopped to look at the map. Realizing that I was going the wrong way, I drove back the other way. c. After driving for a while I got into some stopand-go driving. Once past the tie-up I could speed up again.

Step-by-Step Solution

Answer

Graph (a): ii, Graph (b): iii, Graph (c): i.

Step 1: Identify Story (a) Characteristics

The story involves driving at one speed, getting stopped for a speeding ticket, and then driving at a slower speed. The velocity graph that represents this would start at a certain positive value, drop to zero when stopped (for the ticket), and then return to a lower positive value.

Step 2: Identify Graph for Story (a)

Graph (ii) is the most appropriate for story (a) because it starts with a positive velocity, drops to zero, and then rises to a lower positive velocity.

Step 3: Identify Story (b) Characteristics

The story begins with driving in one direction, stopping to check the map, and then driving in the reverse direction. The velocity graph would start with a positive value, drop to zero (while checking the map), and then continue in the negative direction.

Step 4: Identify Graph for Story (b)

Graph (iii) fits story (b) as it starts with a positive velocity, drops to zero, and then continues with negative velocity, indicating driving in the opposite direction.

Step 5: Identify Story (c) Characteristics

This story involves driving normally, experiencing stop-and-go driving (which involves alternating between zero and positive velocity), and then accelerating to a steady speed. Look for a graph that oscillates between zero and a positive value and then increases steadily.

Step 6: Identify Graph for Story (c)

Graph (i) is the best match for story (c) because it oscillates up and down, representing stop-and-go driving, and then increases gradually, showing acceleration to a higher speed.

Key Concepts

Velocity Analysis

Velocity analysis involves understanding how an object's speed and direction change over time. For a car driving on a road, this involves its movement patterns, such as accelerating, stopping, or moving at constant speed. By analyzing the changes in velocity, we can interpret driving behaviors.

- **Constant Velocity:** When a vehicle moves at a constant speed without changing direction, its velocity graph appears as a horizontal line. - **Positive and Negative Velocity:** A positive velocity indicates movement in the initial direction, while a negative velocity shows the vehicle is moving in the opposite direction. This is crucial when reversing or changing direction.
- **Zero Velocity:** A flat horizontal section at the zero line on a velocity graph indicates that the vehicle is not moving.

Interpreting velocity changes helps understand "why" behind the movements, like speeding up, slowing down, or halting.

Matching Graphs to Stories

Each story corresponds to a specific pattern of velocity on a graph. Understanding how to match them requires knowledge of how driving behavior appears visually.

For example, in story (a), getting a speeding ticket causes a sudden stop, followed by slower driving. This story marries well with a graph starting at a higher velocity, dropping to zero (stop), then moving to slower velocity.

• Story (a): Matches Graph (ii) - Starts fast, stops, and continues slower.
• Story (b): Matches Graph (iii) - Moves forward, stops, then reverses.
• Story (c): Matches Graph (i) - Exhibits stop-and-go motion and speeds up.

You develop intuition about these relationships by noting how key events in a story, like pauses or direction changes, are represented as fluctuations in a graph. With practice, matching becomes a straightforward task, enhancing comprehension of events based on graphical data.

Stop-and-Go Driving

Stop-and-go driving is a common scenario on busy roads where drivers frequently need to start and stop. This driving pattern creates specific features on velocity graphs.

- **Oscillating Velocity:** This involves velocities that repeatedly rise and fall between zero and some positive value, forming a wave-like pattern. We're often stopped at lights or in traffic jams before surging ahead. - **Transition to Steady Speed:** After escaping the congested area, the graph typically trends upward to indicate a return to regular driving speed.
Stop-and-go conditions test driver's patience, requiring frequent braking and acceleration, something crucial for understanding velocity variations on a graph. By being familiar with these patterns, interpreting real-world driving conditions becomes simpler.