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Chapter 3: Problem 4

A runner is practicing on a circular track that is \(300 \mathrm{m}\) in circumference. From the point farthest to the west on the track, he starts off running due north and follows the track as it curves around toward the east. (a) If he runs halfway around the track and stops at the farthest eastern point of the track, what is the distance he traveled? (b) What is his displacement?

Short Answer

Expert verified
Answer: The distance traveled by the runner is 150 meters, and his displacement from the starting point is approximately 95.50 meters.

Step by step solution

01

Find the radius of the circular track

First, we need to find the radius of the circular track. We know that the circumference of a circle is given by the formula: C = 2 * pi * r Where C is the circumference (300 meters in this case) and r is the radius. So, to find the radius, we can rearrange the formula: r = C / (2 * pi) Now, substitute the values: r = 300 / (2 * pi) r ≈ 47.75 meters
02

Calculate the distance traveled by the runner

Since the runner runs halfway around the track, the distance traveled will be half of the circumference. We can calculate this simply by dividing the circumference by 2: Distance traveled = Circumference / 2 Distance traveled = 300 / 2 Distance traveled = 150 meters
03

Calculate the diameter of the circular track

The diameter of a circle is twice its radius. We have already found the radius (r ≈ 47.75 meters), so we can find the diameter as follows: Diameter = 2 * r Diameter ≈ 2 * 47.75 Diameter ≈ 95.50 meters
04

Find the runner's displacement

The displacement of the runner is the straight-line distance from his starting point (farthest west) to his endpoint (farthest east). Since these two points are the endpoints of the diameter of the circular track, the displacement will be equal to the diameter calculated in Step 3. Displacement ≈ 95.50 meters So, the distance traveled by the runner is 150 meters, and his displacement is approximately 95.50 meters.

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