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I'm having trouble with the following problem:

Tom starts running towards a park which is at $800$m from him at speed $20$ m/s. Kate who starts running with Tom at $25$ m/s goes back and forth between park and Tom until Tom reaches the park.

Find total distance covered by Kate.

I calculated the sum until Tom and Kate meet each other the second time, and I get the total distance covered by Kate until that time to be $800+88.9+88.9$.

However after that the calculations become too complex. Is there any other way to do this?

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I did the sum until Tom and Kate meet each other the second time so I get total distance covered by Kate until that to be 800+88.9+88.9. However after that the calculations become too complex is there any other way to do this –  Sudhanshu Aug 4 at 15:57
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Easier: How long does Tom run? How long does Kate run? –  hardmath Aug 4 at 15:59
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Tom runs for 40 s so does that mean Kate covers 25 *40 = 1000 m Is it that simple!! Please tell me if I am doing it right –  Sudhanshu Aug 4 at 16:01
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This is a very similar problem to the famous fly puzzle. See here primepuzzle.com/leeslightest/howfar.html –  Mathmo123 Aug 4 at 16:38
    
1000 meters in 40 seconds has got to be a new world record. –  kasperd Aug 4 at 21:56

2 Answers 2

up vote 9 down vote accepted

The key is that it doesn't matter "where" Kate runs ; we only want to know the distance she runs, and since we know her speed it suffices to know how long she's been running. Since she runs as long as Tom does, we only need to look at Tom's speed and distance. The solution with the geometric series is uselessly over-descriptive but correct too.

Hope that helps,

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Assuming that her speed is constant, with no slowing down or speeding up when she changes direction. –  MattClarke Aug 4 at 23:39
    
@MattClarke : That's the dream. Welcome to physics, where dreams are assumed. –  Patrick Da Silva Aug 5 at 0:41

$$800\frac{25}{20}=1000$$ Kate ran for the same lapse of time as Tom.

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In other words, you don't need to know the time. After any amount of time, Kate has run $25$% ($25/20=1.25 => 25$%) further than Tom. –  Scott Aug 4 at 20:11

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