Question

Can you take a bicycle ride in such a way that the distance you cover is less than the magnitude of your displacement? Give an example if your answer is yes; explain why not if your answer is no.

Short answer

No, because distance is always equal to or greater than displacement.

Step-by-step solution

Understanding the Terms

Distance refers to the total length of the path traveled, regardless of direction, and it is always a non-negative value. Displacement, on the other hand, is a vector quantity that refers to the change in position from the starting point to the endpoint in a straight line, regardless of the path taken.

Analyzing the Problem

The problem asks if the distance covered by a bicycle can be less than the magnitude of the displacement. By definition, distance is the actual path taken, which is always greater than or equal to the straight-line path (i.e., displacement).

Concluding the Analysis

Since any path taken will either match or exceed the straight-line displacement when totaling up the distance, it is impossible for the distance to be less than the displacement. Therefore, there can be no situation where the distance is less than the magnitude of the displacement.

Key concepts

Distance and Displacement

When talking about movement, we often use the terms "distance" and "displacement." Although they might sound similar, they describe different aspects of motion. Distance refers to the total path traveled, no matter where it goes. For example, if you're riding your bicycle around a park, the distance is the length of the whole path you pedal along, even if it's winding or circular.

Displacement, however, is a bit different. It describes how far out of place an object is; it's the shortest path between two points or the straight-line distance from the start to the end. This means it's a vector quantity, and it depends on direction. If you start at the park entrance and make your way around but end back at the entrance, your displacement might be zero, regardless of the distance you pedaled.

• Distance is a scalar, meaning it only has magnitude.
• Displacement is a vector, meaning it has both magnitude and direction.
• Displacement can be zero, even if the distance isn’t.

Understanding these differences is key to solving problems in physics that involve motion.

Vector Analysis

Vector analysis plays a crucial role in comprehending motion problems, like the one given in the bicycle exercise. When we describe displacement as a vector, we mean it has both magnitude and direction. Unlike distance, a vector tells us not just how far but also in which direction.

In any vector analysis, you'll often see components broken down into different directions, usually the horizontal and vertical planes. Think of it like taking a map and placing it flat: the x-axis might represent east-west movement and the y-axis north-south. When analyzing vectors:

• Vectors can be added graphically or mathematically to determine resultant vectors.
• Mathematically, use the Pythagorean theorem to find the magnitude of the resultant vector.
• Angles can help in finding precise displacement direction.

In the context of our bicycle problem, displacement would be the straight line drawn between your starting and ending points on the map. Understanding vectors and how they differ from scalars helps clarify why distance, a scalar, can never be less than displacement, a vector.

Bicycle Motion Analysis

Analyzing bicycle motion offers a practical application of the concepts of distance and displacement. Imagine riding a bicycle around a city block. You start at one corner and pedal all the way around back to your starting point. Here, the distance would be the entire perimeter of the block. However, your displacement is zero, as your ending point is the same as your starting point, forming a vector of length zero.

But what if you rode directly from one corner of the block to the opposite corner in a straight line? Your displacement would now be the direct path between these two points, while your distance is still the same as the displacement.

• Bicycle motion can illustrate clearly how distance and displacement can vary greatly.
• It demonstrates that, while traveling, the actual path (distance) can never be shorter than the straight-line path (displacement).
• In practical scenarios like this, seeing the connection between theoretical concepts and real-life applications becomes much easier.

These concepts together underline why, no matter the route or twists and turns taken during a bicycle ride, the distance covered can never be less than the displacement.