Motorcyclist Circular Motion Problem: Distance And Displacement

Let's dive into a classic physics problem involving circular motion! This problem focuses on a motorcyclist riding around a circular arena, and we need to figure out the distance they travel and their displacement after specific time intervals. It might sound a bit complicated at first, but don't worry, we'll break it down step-by-step. This detailed explanation will help you understand the concepts of distance and displacement in circular motion, and how they differ. We will also delve into how to calculate these values using the information provided in the problem.

Understanding the Problem

Before we jump into calculations, let's make sure we fully grasp the scenario. We have a motorcyclist moving in a circle. The radius of this circle, which is the arena's radius, is 13 meters. The motorcyclist completes one full circle in 8 seconds. This gives us crucial information about the motorcyclist's speed and motion. The key here is to distinguish between distance, which is the total length the motorcyclist travels, and displacement, which is the straight-line distance between the starting and ending points. Think of it this way: if you run a full lap around a track, you've covered a significant distance, but your displacement is zero because you end up back where you started. This distinction is critical in solving this problem. We need to calculate both the distance traveled and the displacement for two different time intervals: 4 seconds and 8 seconds. This will highlight how these two quantities can vary even within the same motion.

Part A: Motion in 4 Seconds

Calculating Distance in 4 Seconds

First, let's tackle the distance traveled in 4 seconds. We know the motorcyclist completes one full circle in 8 seconds. Therefore, in 4 seconds, which is half the time, the motorcyclist will complete half a circle. To find the distance, we need to calculate the length of this half-circle arc. The circumference of the full circle is given by the formula 2πr, where r is the radius. In our case, the radius is 13 meters, so the full circumference is 2 * π * 13 meters, which is approximately 81.68 meters. Since the motorcyclist travels half a circle, the distance covered in 4 seconds is half of the circumference. That means the distance is 81.68 meters / 2, which equals approximately 40.84 meters. So, in 4 seconds, the motorcyclist travels about 40.84 meters along the circular path. Remember, distance is a scalar quantity, meaning it only has magnitude and no direction. We're simply concerned with the total length covered.

Calculating Displacement in 4 Seconds

Now, let's figure out the displacement in 4 seconds. Remember, displacement is a vector quantity, meaning it has both magnitude and direction. It's the shortest straight-line distance between the motorcyclist's starting point and ending point. After 4 seconds, the motorcyclist has traveled half a circle. If we imagine the circle, the starting point and ending point are at opposite ends of the circle, forming a diameter. The displacement is therefore the diameter of the circle. The diameter is twice the radius, so the displacement is 2 * 13 meters, which equals 26 meters. The direction of the displacement would be along the diameter, from the starting point to the ending point. This highlights the difference between distance and displacement. The motorcyclist traveled a considerable distance along the curved path (40.84 meters), but the straight-line distance from start to finish is only 26 meters.

Part B: Motion in 8 Seconds

Calculating Distance in 8 Seconds

Next, let's consider the motion in 8 seconds. This part is a bit simpler because we already know the motorcyclist completes one full circle in 8 seconds. To calculate the distance traveled, we simply need to find the circumference of the circle. As we calculated before, the circumference is 2πr, which is approximately 81.68 meters. So, in 8 seconds, the motorcyclist travels the entire circumference of the circle, covering a distance of about 81.68 meters. This reinforces the concept that distance accumulates along the path of motion, regardless of direction changes. The motorcyclist has traveled a significant length along the circular track.

Calculating Displacement in 8 Seconds

Now, let's determine the displacement after 8 seconds. This is where the concept of displacement truly shines. After completing a full circle, the motorcyclist ends up exactly where they started. Therefore, the straight-line distance between the starting point and the ending point is zero. The displacement is 0 meters. This might seem counterintuitive, especially since the motorcyclist traveled a substantial distance (81.68 meters). However, displacement only considers the net change in position. Since the motorcyclist returned to the initial position, the overall displacement is zero. This perfectly illustrates the difference between distance, which is the total path length, and displacement, which is the change in position.

Key Takeaways

This problem beautifully illustrates the difference between distance and displacement, two fundamental concepts in physics. Distance is the total path length traveled, while displacement is the straight-line distance between the initial and final positions. In circular motion, this distinction becomes particularly important because an object can travel a significant distance while having zero or a small displacement if it returns to its starting point or moves along a curved path. Understanding these concepts is crucial for analyzing more complex motion scenarios. We've also seen how to apply basic formulas like the circumference of a circle to solve physics problems. Remember, always visualize the problem, identify the key information, and choose the appropriate formulas to arrive at the solution. Don't hesitate to break down the problem into smaller, manageable steps, as we did with the 4-second and 8-second intervals. This approach makes even seemingly complex problems much easier to solve.