Displacement Vs. Distance: A Simple Explanation

Hey guys! Ever wondered what the difference is between how far you've walked and how far you've actually moved from your starting point? It's a classic physics problem, and we're going to break it down today. We've got a fun scenario: An individual walks 8 kilometers north and then 6 kilometers east. Let's figure out (A) the magnitude of their displacement and (B) the total distance they traveled.

Understanding Displacement and Distance

Before we dive into the calculations, let's make sure we're all on the same page about what displacement and distance really mean. These two terms are often used interchangeably in everyday conversation, but in physics, they have very distinct meanings.

Distance is the total length of the path traveled. Think of it as the number of steps you've taken or the miles you've clocked on your pedometer. It's a scalar quantity, meaning it only has magnitude (a numerical value). In our scenario, the individual walks 8 km north and then 6 km east, so the total distance is simply the sum of these two lengths.

Displacement, on the other hand, is the shortest distance between the initial and final positions. It's a vector quantity, meaning it has both magnitude and direction. Imagine drawing a straight line from where the person started to where they ended up. That line represents the displacement. Because it's a vector, we need to know not only how long that line is but also in what direction it points. This is where the Pythagorean theorem comes in handy, as we'll see shortly.

Calculating the Magnitude of Displacement (A)

Okay, let's tackle the first part of the problem: finding the magnitude of the individual's displacement. Remember, displacement is the straight-line distance from the starting point to the ending point. Since the person walks north and then east, these two movements are perpendicular to each other, forming a right triangle. This makes our calculation straightforward!

The northward walk of 8 km forms one leg of the right triangle, and the eastward walk of 6 km forms the other leg. The displacement is the hypotenuse (the longest side) of this triangle. We can use the Pythagorean theorem to find the length of the hypotenuse:

• a² + b² = c²

Where:

• a = 8 km (northward distance)
• b = 6 km (eastward distance)
• c = displacement (what we want to find)

Plugging in the values:

• 8² + 6² = c²
• 64 + 36 = c²
• 100 = c²

To find c, we take the square root of both sides:

• √100 = c
• c = 10 km

So, the magnitude of the displacement is 10 kilometers. This means the person is 10 km away from their starting point in a straight line. But remember, displacement is a vector, so we also need to consider the direction. We'll get to that in a bit, but for now, we've answered the first part of the question.

Determining the Total Distance Traveled (B)

Now for the easier part: calculating the total distance traveled. As we discussed earlier, distance is the total length of the path taken. In this case, the individual walked 8 km north and then 6 km east. To find the total distance, we simply add these two distances together.

Total distance = Distance north + Distance east Total distance = 8 km + 6 km Total distance = 14 km

Therefore, the individual traveled a total distance of 14 kilometers. Notice that this is different from the displacement (10 km). This is because displacement is a straight-line measure, while distance accounts for the entire path, even if it's not a straight line.

The Importance of Direction in Displacement

We've calculated the magnitude of the displacement, which is 10 km. But to fully describe the displacement, we also need to specify its direction. We can do this by finding the angle the displacement vector makes with the horizontal (eastward) direction.

Imagine drawing a line from the starting point to the ending point. This line is the displacement vector. It forms an angle with the eastward direction. We can use trigonometry to find this angle. Specifically, we can use the tangent function:

tan(θ) = opposite / adjacent

In our right triangle:

• The opposite side is the northward distance (8 km).
• The adjacent side is the eastward distance (6 km).
• θ is the angle we want to find.

So:

tan(θ) = 8 km / 6 km tan(θ) = 1.333

To find θ, we take the inverse tangent (arctan) of 1.333:

θ = arctan(1.333) θ ≈ 53.1 degrees

Therefore, the direction of the displacement is approximately 53.1 degrees north of east. This means the person's final position is 10 km away from their starting point in a direction that's about 53.1 degrees towards the north from the east direction.

Real-World Applications of Displacement and Distance

Understanding the difference between displacement and distance isn't just a theoretical exercise; it has many real-world applications. For example, consider a delivery truck making multiple stops in a city. The total distance the truck travels affects fuel consumption and wear and tear on the vehicle. The displacement, on the other hand, might be more relevant for planning the most efficient route back to the depot.

In sports, the distance a runner covers in a marathon is important for tracking their overall effort. However, their displacement (the straight-line distance from the starting line to the finish line) is what ultimately determines their race result.

Navigation systems also use concepts of displacement and distance. GPS devices calculate your position and displacement to provide accurate directions, while also tracking the total distance you've traveled.

Key Takeaways

Let's recap the key concepts we've covered:

• Distance is the total length of the path traveled, a scalar quantity.
• Displacement is the shortest distance between the initial and final positions, a vector quantity with both magnitude and direction.
• We can use the Pythagorean theorem to find the magnitude of displacement when movements are perpendicular.
• Trigonometry helps us determine the direction of displacement.
• Displacement and distance have different real-world applications, from navigation to sports.

So, next time you're tracking your steps or planning a trip, remember the difference between distance and displacement. It's a fundamental concept in physics that helps us understand motion in a more complete way. Keep exploring, guys, and stay curious!

Final Answer

A) The magnitude of the displacement is 10 km. B) The total distance traveled is 14 km.