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Traveling in a car: Make graphs of location and velocity for each of the following driving events. In each case, assume that the car leaves from home moving west down a straight road and that position is given as the distance west from home. a. A vacation: Being eager to begin your overdue vacation, you set your cruise control and drive faster than you should to the airport. You park your car there and get on an airplane to Spain. When you fly back 2 weeks later, you are tired and drive at a leisurely pace back home. (Note: Here we are talking about location of your car, not of the airplane.) b. On a country road: A car driving down a country road encounters a deer. The driver slams on the brakes and the deer runs away. The journey is cautiously resumed. c. At the movies: In a movie chase scene, our hero is driving his car rapidly toward the bad guys. When the danger is spotted, he does a Hollywood 180-degree turn and speeds off in the opposite direction.

Step by step solution

01

- Understanding the Scenario

To solve the problem, we first need to understand that you need to create position and velocity graphs for three distinct scenarios presented. Each situation involves changes in the car's position and velocity over time.

02

- Analyzing Event a: A vacation

In this event, the car travels quickly from home to the airport, stays at the airport, and then returns home slowly. The position graph should show a rapid increase in the westward distance at first, a horizontal line when the car is stationary at the airport, and a gradual decline as the car slowly returns. The velocity graph will show a constant high velocity, followed by zero velocity when parked, and a constant lower velocity back home.

03

- Analyzing Event b: On a country road

In this situation, the car's position increases as it moves down the road until it suddenly stops when the brakes are applied. After pausing (position remains constant), it resumes moving forward slowly. The position graph will show an initial increase, a flat segment during the stop, and then a gradual rise. The velocity graph will depict a positive constant velocity until a sudden drop to zero, followed by a bounce back to a lower positive constant velocity.

04

- Analyzing Event c: At the movies

Here, the car's position initially increases rapidly while approaching the bad guys, then ceases when the car stops to turn, and finally decreases as the car accelerates in the opposite direction. The position graph should show a steep rise, a brief pause during the turn, and a subsequent rapid decrease as it drives back. The velocity graph will reflect a high positive velocity, followed by zero velocity during the turn, and then a high negative velocity reflecting motion in the opposite direction.

05

- Drawing the Graphs

For each event, sketch two graphs (position-time and velocity-time). Start by marking the axes. For position-time, position (distance west) goes on the y-axis and time on the x-axis. For velocity-time, velocity is on the y-axis and time on the x-axis. Use the analyzed behavior to draw each graph, ensuring to capture the changes in motion and stationary phases accurately.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Position-time graph

A position-time graph allows us to visualize how the location of an object changes over a period of time. This kind of graph is useful because it provides a snapshot of an object's journey and helps us understand its motion more clearly.

• On this graph, the horizontal axis (x-axis) represents time, typically measured in seconds, minutes, or hours.
• The vertical axis (y-axis) represents the position or distance of the object from a chosen starting point. In our case, this is the distance west from home.

By observing the slope of the line on a position-time graph, we can determine how fast an object is moving.

• A steeper slope indicates a faster speed.
• A flat horizontal line means the object has stopped moving.
• If the line slopes downward, the object is returning to the starting point or moving in the opposite direction to the original motion.

With this understanding, we can plot the car's journey in each scenario, showing changes in position as it moves to the airport, stops, and returns home, or how it handles unexpected events like a sudden stop or change in direction.

Velocity-time graph

A velocity-time graph is a powerful tool to understand how an object's speed and direction change over time. These graphs are helpful to determine acceleration and steady-state speeds during an activity.

• The x-axis represents time, just like in the position-time graph.
• The y-axis now shows velocity, which is the speed of the object in a specific direction. In physics, velocity is a vector, which means it has both magnitude and direction.

A deeper look into the velocity-time graph assists in comprehending the motion:

• A flat line above the x-axis shows constant velocity in a positive direction.
• A flat line at zero indicates the object is at rest.
• When the line slopes downward and crosses the x-axis, the object slows down, stops, or starts moving in the opposite direction.
• A steeper line indicates faster acceleration or deceleration.

By constructing a velocity-time graph for each scenario, we can pinpoint when the driver speeds up, brakes suddenly, or reverses direction, capturing the essence of motion for each driving event.

Position and velocity changes

Understanding how position and velocity change during motion can deepen our insight into the dynamics of how an object, like a car, behaves under different circumstances.

• In event a, when the car races towards the airport and later cruises back, both position and velocity graphs showcase the initial rapid increase in movement, a stationary phase, and then a gradual slowdown in speed as it returns home.
• During event b, encountering a deer results in a sudden change in both position (the car stops) and velocity (velocity drops to zero abruptly).
• Event c illustrates how a movie-style chase involves rapid changes: the car speeds towards the threat, halts for a turn, and switches to a chase mode in the opposite direction. This is captured as rapid position shifts and reversed velocity.

These scenarios highlight the interplay between an object's speed, direction, and position shifts over various timeframes. By analyzing events descriptively as well as graphically, students can gain a more holistic understanding of motion and the physics concepts that define it.