Problem 4 - An Introduction to Physical Science

Question

A jogger jogs around a circular track with a diameter of \(300 \mathrm{m}\) in 10 min. What was the jogger's average speed in \(\mathrm{m} / \mathrm{s}\) ? What is the magnitude of the jogger's average velocity?

Step-by-Step Solution

Answer

The jogger's average speed is approximately \(1.57 \mathrm{m/s}\) and the average velocity is \(0 \mathrm{m/s}\).

Step 1: Calculate the Circumference of the Track

To find the jogger's average speed, we first need to calculate the circumference of the circular track. The formula for the circumference \(C\) of a circle is \(C = \pi d\), where \(d\) is the diameter of the circle. Given that the diameter of the track is \(300 \mathrm{m}\), the circumference is computed as follows: \[ C = \pi \times 300 \approx 942 \mathrm{m}. \]

Step 2: Determine the Jogger's Average Speed

Average speed is defined as the total distance traveled divided by the total time taken. The jogger travels a distance equal to the circumference of the track in 10 minutes. First, convert 10 minutes to seconds: \(10 \times 60 = 600\) seconds. The average speed \(v\) is then: \[ v = \frac{942 \mathrm{m}}{600 \mathrm{s}} \approx 1.57 \mathrm{m/s}. \]

Step 3: Calculate the Magnitude of the Jogger's Average Velocity

Average velocity is the displacement divided by the time taken. Since the jogger returns to their starting point after completing one lap, the displacement is zero. Therefore, the magnitude of the average velocity is: \(0 \mathrm{m/s}.\)

Key Concepts

Circular Track Calculations

When dealing with circular tracks, one of the most crucial concepts is understanding how to calculate the track's circumference. The circumference represents the complete distance around the track once. To find it, you simply use the formula \( C = \pi d \), where \( d \) is the diameter of the track. In this exercise, the diameter is given as \( 300 \text{ m} \). So, the circumference \( C \) can be calculated as \( \pi \times 300 \approx 942 \text{ m} \). This value is key because it's the total distance the jogger travels on a complete lap.
Remember, the circumference is directly proportional to the diameter, meaning bigger tracks with larger diameters will naturally have greater circumferences. Knowing how to calculate this is essential in scenarios involving movement around circular paths, as it forms the basis for understanding concepts like average speed.

Average Velocity

While average speed and average velocity may sound similar, they differ quite significantly in physics. Average speed is all about the total distance covered over time. However, average velocity considers displacement - the direct line distance from the start point to the end point - over time.

In circular motion, especially, this distinction becomes very apparent. If you complete a full lap around a circular track, your starting and ending points will be the same, which means your displacement is zero. Consequently, this makes the average velocity zero as well, despite covering a considerable distance.

• This difference highlights a common confusion in physics where covering distance doesn't always equate to displacement.
• For situations involving repetitive circular paths, understanding that the average velocity accounts for only the start and stop point really helps clarify why, despite traveling a long way, the final average velocity could be zero.

Understanding this concept not only helps solve problems involving motion but also builds a clearer picture of how motion is tracked in different ways.

Displacement in Circular Motion

Displacement in circular motion can often be misunderstood, but it's a central part of calculating average velocity as shown in the exercise. Displacement is defined as the shortest path between the initial and final points.

For any circular track, if you start and finish at the same point, your net displacement is zero.

• This principle is crucial for understanding why the average velocity for a full lap around a circular track is zero.
• Understanding displacement emphasizes the concept that motion over time considers not just how far you go, but where you end up relative to where you started.

While this might seem straightforward in cases where you return to the origin, it becomes more complex when different fractions of the track are involved. In those scenarios, only the direct line distance between the initial and current position will count, significantly altering the average velocity calculations. This highlights that the path traveled might be circular or winding, but the displacement effectively simplifies it to the direct route.