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A question from my homework: Two MSU cross-country runners are 440 yards apart and are...

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A question from my homework: Two MSU cross-country runners are 440 yards apart and are running towards each other, one at 8mph and the other at 10 mph. In how many seconds will they meet? How do I set this up?

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First, we note that 440 yards is equivalent to 0.25 miles. 

To answer the problem, we first try to visualize what's happening.

There are two runners, A and B. The speed of A is 8mph, that of B is 10 mph. A and B are standing on opposite ends of a track that is 0.25 miles long. They are running towards each other. Now, since they are running towards each other, they will definitely meet. The point of meeting is important -- it will be at the same point (of course) and they would have ran after the same amount of time (of course, assuming they started at the same time).

A --------x-------------------B

In the illustration above, x would be the meeting point. Obviously, B would have covered more distance since he has a greater speed. Assuming A covered a distance of x miles, B would have covered a distance of 0.25 - x miles (since the total distance is 0.25 miles. Also, note that we could have assigned x to B and 0.25-x to A but that would not affect the answer). Then, we let the total time lapsed equal to y. The time is equal for both runners.

To summarize our variables:

Speed of A: 8 mph

Speed of B: 10 mph

Distance Travelled by A: x miles

Distance Travelled by B: 0.25 - x miles

Time A Ran: y

Time B Ran: y

Next, we note that speed is simply distance over time. From the given, we can obtain two equations:

`x/y = 8`

`(0.25 - x )/ y = 10`

The first equation is simply `x = 8y` . The second equation reduces to `0.25 - x = 10y` , which, using equation one, can be easily solved as: `0.25 - 8y = 10y rightarrow 0.25 = 18y` . This means that `y = 0.25/18 =0.013888889 hrs` , which is just 50 seconds.` `

Hence, the runners will meet after 50 seconds.


To check, simply solve for the distance travelled by both after 50secs or 0.01389 hours.

Distance (A) = 8 mph * 0.01389 h = 0.11112 miles

Distance (B) = 10 mph * 0.01389 h = 0.1389 miles

Total (A+B) = 0.11112 + 0.1389 = 0.25 miles

The answer must be correct. The total distance is 0.25 miles, and hence the two must have met. Also, B travelled more distance since s/he is faster. If they had the same speed, they would simply meet at the center.

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