Question

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

Short answer

Yes, walking on a curved path or winding route results in greater distance than displacement; example: walking 3 meters north and then 4 meters east has greater distance than displacement.

Step-by-step solution

Understanding the Terms

First, let's clarify the terms 'distance' and 'displacement'. - **Distance** is the total path length traveled, regardless of direction. - **Displacement** is the straight line between the start and end points, inclusive of direction.

Walking in a Straight Line

Consider first if you walk in a straight line from point A to point B. In this case, the distance you cover and the magnitude of your displacement will be equal, as both measure the direct path from A to B.

Walking on a Curved Path

Let's think about walking on a path that isn't straight. For example, if you make a circular lap around a park. When you return to your starting point after one complete circle, the displacement (the straight line distance between your starting point and end point) is zero, while the distance walked equals the circumference of the circle.

Example of Greater Distance

Here's a practical example: If you walk 3 meters north, then 4 meters east (a right-angled path), your distance walked is 7 meters. However, the displacement, calculated using the Pythagorean theorem, is \( \sqrt{3^2 + 4^2} = 5 \) meters.

Conclusion

When walking on curved or non-linear paths, the distance traveled is typically greater than the magnitude of displacement, unless you're moving straight from point A to B along a straight line.

Key concepts

Distance

Distance in physics refers to the total length of the path traveled by an object, without considering its direction.

Imagine walking in a park and taking several turns; the distance is the sum of all the steps you take, no matter how twisty the path.

This means distance considers every inch of your journey, even if it loops back or winds around.

• Distance measures the entire journey taken.
• It's a scalar quantity, meaning it has magnitude but no directional component.
• No matter how you twist and turn, each step adds to the total distance.

Understanding distance is about comprehending the full path, making it often longer than simply measuring the start to finish line.

Displacement

Displacement is a different kind of measure. It focuses on the straight-line path between your starting point and destination.

Displacement considers the shortest path possible, making it like a shortcut through your multiple twists and turns.

• Displacement includes direction, as it's a vector quantity.
• It always points from the start to the end position.
• If you return to your starting point, your displacement is zero.

This attribute is crucial when studying motion because it gives a more direct and efficient understanding of where an object ends up relative to where it began.

Pythagorean Theorem

The Pythagorean Theorem is a mathematical principle that helps calculate the displacement when movement includes right-angled turns.

This theorem applies mostly in scenarios involving perpendicular components, like moving north and then east. If you know the lengths on each axis, it helps find the direct line or displacement.

Here's how it works:

• It's used when you travel in two directions at right angles to each other.
• The formula is written as: \(c^2 = a^2 + b^2\) where \(c\) is the hypotenuse or displacement.
• In our example, moving 3 meters north and 4 meters east results in a 5-meter displacement, calculated as \(\sqrt{3^2 + 4^2}\).

This makes it a handy tool in figuring out the direct path one's end-to-end movement represents.

Curved Path

A curved path presents a unique situation where your distance traveled can greatly exceed your displacement.

Consider walking in a perfect circle. Once you come back to where you started, the displacement is zero because there's no "straight-line distance" when start and end points match.

• Walking a full circle gives high total distance.
• Displacement resets to zero if you loop back to the start.
• This is why sports tracks or walking paths in parks often have high travel distances but low displacement.

Understanding this helps differentiate between how much one moves and just the effective "crow flies" distance covered.

Straight Line Motion

Straight line motion implies moving from one point to another with no detours or turns. Here, distance equals displacement since there's no deviation from the path.

Think about heading directly across a playground without wandering to sides. Every step directly contributes to both distance and displacement.

• Distance equals displacement in perfect straight line motion.
• This motion is simplest to analyze due to no deviations.
• Useful for understanding basic physics concepts like velocity and acceleration.

When motion follows a straight line perfectly, both distance and displacement are easy to calculate, showing how streamlined motion helps get straight to points.