- #1
Bagbo
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On project Eureka the following problem was posted:
You and a friend like running, but your friend is much faster than you. So you can run together, you two have arranged the following system. You both start out at the beginning of a path and run at constant speeds. Your friend runs to the end of the path, turns around, and then meets you along the path. Your friend then repeats this process two more times (running to the end of the path, turning around, and then meeting you). Your first meeting happens after one hour; the third meeting happens after two hours.
What portion of the path do you cover in the first hour?
The accepted solution 38.2 and a description was available from another site
The 'solution' is described as follows:
""Alas, no one got the solution... Here's how I did the problem. Use units of length so that the length of the path is 1. Let s1 be your speed (in units per hour), while s2 is your friend's speed. During the first hour, both of you together run the path twice. So we know that s1 + s2 = 2 and that you are at position s1, leaving you a distance of 1 - s1 to the end of the path.
Between the first and second meeting, you and your friend will cover that distance twice. If that takes time t, we know that (s1 + s2) * t = 2t = 2(1 - s1). So the time to the second meeting is t = 1 - s1. You are now at position s1 + s1*(1 - s1), leaving a distance of 1 - 2s1 + (s1)2 to run.
Between the second and third meeting, you and your friend will cover that distance twice. Again, if that takes time t, we know that 2t = 2(1 - 2s1 + (s1)2). So the time to the third meeting is t = 1 - 2s1 + (s1)2. Thus the total time run so far is 1 (for the first meeting) plus 1 - s1 (for the second meeting) plus 1 - 2s1 + (s1)2 (for the third meeting). Setting this equal to 2 hours and solving gives us s1 = (3 - sqrt(5))/2 or about 38.2%.""
I don't think this is correct. There is no justification for claiming the (1 - s1) distance will be covered twice by the joggers between the first and second meeting. What will be covered by the joggers is 2s1 + 2s1*t hence t = 2s1(1 + t)/2 = s1(1 + t). Hence t = s1/(1 - s1). Since the second meeting must be between 1 and 2 hours and the joggers must meet a third time after 2 - 1 - t , I believe ANY s1 such that 0 < s1 < 0.5 is a valid answer including 38.2.
Is there something I am not seeing?
You and a friend like running, but your friend is much faster than you. So you can run together, you two have arranged the following system. You both start out at the beginning of a path and run at constant speeds. Your friend runs to the end of the path, turns around, and then meets you along the path. Your friend then repeats this process two more times (running to the end of the path, turning around, and then meeting you). Your first meeting happens after one hour; the third meeting happens after two hours.
What portion of the path do you cover in the first hour?
The accepted solution 38.2 and a description was available from another site
The 'solution' is described as follows:
""Alas, no one got the solution... Here's how I did the problem. Use units of length so that the length of the path is 1. Let s1 be your speed (in units per hour), while s2 is your friend's speed. During the first hour, both of you together run the path twice. So we know that s1 + s2 = 2 and that you are at position s1, leaving you a distance of 1 - s1 to the end of the path.
Between the first and second meeting, you and your friend will cover that distance twice. If that takes time t, we know that (s1 + s2) * t = 2t = 2(1 - s1). So the time to the second meeting is t = 1 - s1. You are now at position s1 + s1*(1 - s1), leaving a distance of 1 - 2s1 + (s1)2 to run.
Between the second and third meeting, you and your friend will cover that distance twice. Again, if that takes time t, we know that 2t = 2(1 - 2s1 + (s1)2). So the time to the third meeting is t = 1 - 2s1 + (s1)2. Thus the total time run so far is 1 (for the first meeting) plus 1 - s1 (for the second meeting) plus 1 - 2s1 + (s1)2 (for the third meeting). Setting this equal to 2 hours and solving gives us s1 = (3 - sqrt(5))/2 or about 38.2%.""
I don't think this is correct. There is no justification for claiming the (1 - s1) distance will be covered twice by the joggers between the first and second meeting. What will be covered by the joggers is 2s1 + 2s1*t hence t = 2s1(1 + t)/2 = s1(1 + t). Hence t = s1/(1 - s1). Since the second meeting must be between 1 and 2 hours and the joggers must meet a third time after 2 - 1 - t , I believe ANY s1 such that 0 < s1 < 0.5 is a valid answer including 38.2.
Is there something I am not seeing?