Chasing Cheetahs

A really helpful way to keep things straight when converting units is to treat the units like they were variables.  Using this method to convert between feet and inches, you only need to know that 1 ft= 12 in, and you will never be confused about whether to multiply or divide. To convert 5 ft to inches it would go like this.

Since 12in=1ft, (12in)/(1ft)=1, and since any number times 1 is itself, this won't change the size, just the units. The way to figure out whether to multiply by (12in)/(1ft) or (1ft)/(12in) is simply to do it in such a way that the units cancel out, and check it by doing the canceling. Since ft. is in the numerator we need to have it in the denominator and in. in the numerator so that the ft. will cancel out and the in. will stay. This method of including the units in the calculation and dealing with them as if they were variables is called dimensional analysis.

Dimensional analysis is particularly nice when you have compound units like mi/hr or ft/sec. These are compound units because they are obtained doing a computation and letting the units come right along for the ride. Using speed=distance/time when the distance is in miles and the time is in hours we get the speed in mi/hr, but when the distance is in ft and the time is in seconds we get ft/sec. For conversions these kind of units work just like they were the divisions that they come from, so to convert 60 mi/hr to ft/sec it would go like this.

We put the miles downstairs and the feet upstairs, because the original miles is upstairs and we want to cancel it out, but we put the hours upstairs and the minutes downstairs, because the original hours are downstairs and we want the new minutes unit to be downstairs so that we can then replace it with seconds by the same procedure. For an in depth account of dimensional analysis see Stan Brown's article How to Convert Units of Measure. But if Stan's examples aren't enough to keep you busy, here are some more with an interesting twist to them, quite literally in that they involve turning the units upside down.

What if you want to convert from min/mi or min/km to mi/hr or km/hr? Runners and athletics spectators are sometimes interested in this, because running speeds are often given in the first kind of unit, which is called pace. For example a few days ago I wanted to set a treadmill to 7 min/mi pace, but the treadmill only read km/hr. With a little bit of care you can apply dimensional analysis to this problem too. You just have to realize that these units are slightly different kinds of units. They are reciprocal (upside down) to eachother. Pace is time/distance and speed is distance /time, so to convert pace to speed you have to first invert it or turn it upside down, and if you are going to turn the unit upside down, you have to turn the number upside down as well, so 7 min/mi = 1/7 mi/min and now we can do the conversion.

to the nearest tenth, which is all the treadmill will take.

To figure out how to reverse the process we can look at another computation that I once made having to do with treadmills. Most of the treadmills at the Lakenheath gym only allow you to go up to 14.4 k/hr. How fast is that in min/mile? Again we invert to get k/hr and then do the conversion to mi/hr.

But here I am faced with another problem that dimensional analysis can also solve. I didn't want it in decimal minutes. I wanted it in minutes and seconds that old sexigisimal system that we inherited from the ancient Babylonians and still use for expressing time. To convert to minutes and seconds we have to convert the decimal part into seconds, but again for that all we need to know is how many seconds there are in a minute. to the nearest second, so the final answer is 6:42 min/mi pace is the fastest that the treadmill will allow you to go.

For another application, when I was a child I saw in the World Book Encyclopedia a chart comparing speed of various animals. The speed for man was given as 20 mi/hr. I want to know how that speed compares with what the top sprinters do. One way to look at this is to compute how many seconds/hundred meter that is. To do that we could convert it to sec/m and then multiply by 100. First we convert it to hr/mi by my upside down rule. 20 mi/hr=1/20 hr/mi. Then dimensional analysis can give us the answer like this.

Multiplying by 100 we get that this is 11.187 for a hundred meter sprint, fast but plenty can run faster than that, particularly when you take into account start time and the fact that nobody can run at top speed for a whole 100m.

Another way to compare would be to convert the world record 100m time to mi/hr. Maurice Greene has run 100m in 9.79 seconds. To convert that to mi/hr it works well to simply compute the speed directly from the distance and time in m/sec carrying along the units as is standard in dimensional analysis and then multiply by the appropriate unit fractions in order to convert to mi/hr.

Wow! I'd hate to run into something at that speed. But even he couldn't get away from a cheetah. Cheetahs can run 75 mi/hr!

Well, the top sprinters can run faster than that 20 mi/hr, but I certainly can't. The best I have ever done for 100m is 16 sec. Let's see, that's 100m/4sec. Converting to mi/hr gives me

much slower than 20 mi/hr, but I can go a little faster over a shorter distance. I managed to run 30m from a rolling start once in 4 seconds. That's 30 m/4 sec or Well, that sounds a little faster, but I think I'd better stay away from cheetahs.

Student Exercise: What would a cheetah's 100m time  be if it could maintain 75 mi/hr for the whole distance?

Note of interest about cheetahs: One reason that cheetahs can run so fast is that they have built in spikes. Unlike other cats, they don't retract their claws when they run, instead they use them for spikes. I learned this from a fascinating program on Radio 4 about the biomechanics of various predators.

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