Swimmers
Two friends, Alan and Brian, dive in simultaneously from opposite ends of a non-standard swimming pool and swim at constant, but different, speeds.
If they first cross at a distance 30m from Alan's starting end and next cross at a distance 20m from Brian's starting end, how long is this particular pool and where will they next cross ?
Can you generalise this finding?
There are two types of crossing : head-on, as in the illustration, and overtaking. You might need to consider these separately.-
Why wasn't the second crossing in the same place as the first?
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If the swimmers can keep this going indefinitely will they have eventually had a crossing point everywhere along the length of the pool?
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What happens if they both swim twice as fast?
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What would happen if the pool was twice as long?
At their first crossing A has swum 30 and B has swum x - 30 , where x is the pool length.
At their second crossing A has now swum x + 20 in all and B has swum 2x - 20
There respective distances will keep in a fixed ratio because their speeds are in a fixed ratio.
This equation expresses that
x is zero or 70, and a pool of length zero is not our interest here so 70 m is the pool's length.
Here's another line of reasoning :
At first crossing A and B have swum one length between them, and A swam 30m of that length.
At second crossing A and B have swum two more lengths between them, so A will have swum 60m of those two lengths.
A has swum 90m in all, and that is one whole length plus a further 20m . So the pool is 70m long.
Why do this problem:
At first glance it may seem surprising that the two given distances fix the length of the pool. This context will also stimulate visualisation, as students 'picture' the motion of each swimmer and then watch for the moments when they cross.
Possible approach :
It is important that the context is well understood so that visualisation can be productive. To help this, bring two volunteer walkers to the front of the classroom. Place them on opposite sides and direct one walker to move a little more briskly than normal and the other to walk a little more slowly.Set them off and at their first crossing pause them and mark the position. Set them off again and catch the position of the second crossing similarly.
Next involve some numbers and some calculation to connect some arithmetic with the context that students are starting to visualise.
Invent a pool length (say 100m) and two swimming speeds (2m/s and 2.5 m/s are good). Invite students to calculate the crossings that happen with these values.
The values can then be altered, using calculators when required, until students have a good feel for this problem context.
Use the questions and additional activity below to draw students into this simple but potentially rich context of enquiry and problem solving.
Key questions :
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Why wasn't the second crossing in the same place as the first?
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If the swimmers can keep this going indefinitely will they have eventually had a crossing point everywhere along the length of the pool?
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What happens if they both swim twice as fast?
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What would happen if the pool was twice as long?
Possible extension :
- What, in general, is the connection between the two given distances and the length of the pool?
- What would be the effect, if any, of having both distances given from the same end?
- Do any patterns emerge as you consider the sequence of crossing places or crossing times? And if so can you account for these?
- Think about how you have visualised these swimmers going up and down the pool, what have you pictured in your mind to help you solve this problem? Can you connect any of your results to the questions above into that visualisation?
- Do any of the results prompt you to look for new ways to visualise this general situation?