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Chapter 2: Problem 64

In the problems, please assume the free-fall acceleration $g=9.80 \mathrm{m} / \mathrm{s}^{2}$ unless a more precise value is given in the problem statement. Ignore air resistance. To pass a physical fitness test, Marcella must run \(1000 \mathrm{m}\) at an average speed of \(4.00 \mathrm{m} / \mathrm{s} .\) She runs the first $500 \mathrm{m}\( at an average of \)4.20 \mathrm{m} / \mathrm{s} .$ (a) How much time does she have to run the last \(500 \mathrm{m} ?\) (b) What should be her average speed over the last \(500 \mathrm{m}\) in order to finish with an overall average speed of \(4.00 \mathrm{m} / \mathrm{s} ?\)

Short Answer

Expert verified
Answer: (a) Marcella has 130.95 seconds to run the last 500 meters, and (b) she should have an average speed of approximately 3.82 m/s for the last 500 meters to achieve the desired overall average speed.

Step by step solution

01

Calculate the time taken for the first 500m

We are given that Marcella runs the first 500m at an average speed of 4.20 m/s. We will use the formula for average speed to find the time taken for the first 500m. Average speed = Distance / Time 4.20 m/s = 500m / Time Time = 500m / 4.20 m/s = 119.05 s
02

Calculate the total time for 1000m

Since Marcella needs to run the full 1000m at an average speed of 4.00 m/s, we can find the total time for the run using the same formula: 4.00 m/s = 1000m /Total Time Total Time = 1000m / 4.00 m/s = 250.00 s
03

Calculate the time allowed for the last 500m

Now that we know the total time and the time taken for the first 500m, we can find the time allowed for the last 500m: Time allowed for the last 500m = Total Time - Time taken for the first 500m Time allowed for the last 500m = 250.00 s - 119.05 s = 130.95 s
04

Calculate the required average speed for the last 500m

To find the required average speed for the last 500m, we will use the formula for average speed: Required average speed = Distance / Time Required average speed = 500m / 130.95 s ≈ 3.82 m/s So, the answers are: (a) Marcella has 130.95 seconds to run the last 500 meters, and (b) She should have an average speed of approximately 3.82 m/s during the last 500 meters to finish the race with the desired overall average speed of 4.00 m/s.

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