Question

Describe a scenario where an object's average speed is a large number, but the magnitude of the average velocity is not a large number.

Short answer

A car running in circles at high speed has a large average speed but a small average velocity.

Step-by-step solution

Understanding Speed and Velocity

First, recall the definitions: - Speed is a scalar quantity representing how fast an object travels over a distance in a given period, without considering direction. - Velocity is a vector quantity that includes both speed and direction. Average velocity is the total displacement divided by the total time. Thus, it's possible for an object to travel a great distance (large average speed) but end up near its starting point (small displacement, thus small average velocity).

Choosing the Scenario - Circular Track

Imagine a car moving in a circular track with a radius of 100 meters. The car completes 10 laps in 1000 seconds. Each lap covers a circumferential distance of approximately 628.3 meters. This illustrates how substantial distance is covered over time, but the initial and final positions are the same or very close, making displacement very small or almost zero.

Calculating Average Speed

The average speed is calculated by taking the total distance traveled and dividing by the time taken. Here, the total distance is 10 laps × 628.3 meters/lap = 6283 meters. The time taken is 1000 seconds. Thus, the average speed is:\[\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{6283 \text{ meters}}{1000 \text{ seconds}} = 6.283 \text{ m/s}\]

Calculating Average Velocity

Average velocity is found by dividing the total displacement by the total time. Since the car ends its motion very close to the starting point after 10 laps, the total displacement is approximately zero. Hence, the average velocity is:\[\text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}} \approx \frac{0 \text{ meters}}{1000 \text{ seconds}} = 0 \text{ m/s}\]

Comparing Both Quantities

The scenario depicts a high average speed due to the large distance covered, while the average velocity is negligible because the net change in position is small. This effectively illustrates how average speed can be large even when average velocity is not, due to the path taken by the object.

Key concepts

Scalar and Vector Quantities

Understanding the difference between scalar and vector quantities is fundamental in physics. Scalars are quantities that are described only by magnitude. They do not have a direction. Common examples include:

• Speed
• Distance
• Time
• Mass

Vectors, on the other hand, include both magnitude and direction, making them crucial for describing more complex physical processes. Velocity, for example, is a vector since it describes how fast an object is moving and in which direction. This difference becomes significant in scenarios like circular motion, where speed remains constant, but direction changes continuously. Thus, understanding how vectors can change while maintaining magnitude is key to analyzing many physics problems.

Circular Motion

Circular motion is a fascinating concept in physics that offers insights into how objects move along a circular path. It's essential to note that while an object moving in a circle at constant speed has uniform circular motion, its velocity is actually not constant. This is because velocity is a vector, and its direction is always changing even if the speed remains the same. For instance, consider a car traveling on a circular track:

• The car's speed might be steady, but its direction changes continuously as it moves around the track.
• This means the car is undergoing continuous acceleration despite moving at constant speed.

Understanding circular motion helps explain why objects can have a high average speed, yet a low average velocity, especially when their starting and ending points are the same.

Displacement and Distance

In physics, distance and displacement are two crucial concepts often confused. Distance refers to the total path length traveled by an object, ignoring its starting and finishing positions. It's a scalar quantity, meaning only magnitude is considered.
Displacement, however, is a vector quantity. It represents the change in position of an object and considers the straight line distance from start to finish, taking direction into account. This is why in our circular motion example, although the car travels a considerable distance, its displacement after a full circuit is almost zero. Understanding these differences is crucial for solving motion analysis problems where path and position must be considered.

Physics Problems

Physics problems, particularly those related to motion, often require comprehensive understanding of the relationships between different physical quantities. By breaking down problems into smaller, understandable parts, students can effectively tackle even complex scenarios. Common tips that can help include:

• Identifying what quantities are given and what needs to be found.
• Diagrams can help visualize scenarios, especially in problems involving vectors and motion.
• Reviewing equations and understanding their derivations to apply them correctly.

In our earlier example, recognizing the car’s path as circular was crucial for determining both its speed and velocity, providing a complete analysis of its motion.

Motion Analysis

Motion analysis involves dissecting how objects move from one point to another and is fundamental in physics. This analysis includes understanding various physical quantities and how they relate:

• Speed and velocity: Even if an object moves rapidly over a long period, its velocity can still be low if it returns to its start point.
• Acceleration: Crucial in scenarios where velocity changes, such as in circular motion.
• Graphs and equations are useful tools for visualizing and predicting motion patterns.

In essence, motion analysis is about understanding and predicting the behavior of moving objects. It allows physicists to systematically approach problems and derive meaningful conclusions about the physical world.