When I went to the British museum as a part of my holidays recently, I saw there really interesting (someone would say weird) clock. And a ball which was counting there minutes by running back and forth on a special desk.

There was an information about the length of the ball's journey each year, which was 2,500 miles. That's pretty impressive. And even more impressive is - how did they calculate the length?

Well, that's pretty simple, actually, but still - it's impressive, how could we get interesting facts by using maths.

So the ball runs each year 2,500 miles on the funny desk. Using simple physics (I know, that's even worse than maths...) formula saying that distance is a product of speed and time we can calculate this impressive 2,500 miles easily: when knowing the time - that was 30 seconds according to the information written next to the clock - and the length of the corridor for the ball... That would give us the speed (length per time) and we choose a year as a time for our formula.

But are we not missing something? Oh yes, the actual length of the ball-corridor...

Let's have a look:

s = v * t

distance = speed * time

(2,500 miles - let's pretend we don't know this yet) ? = (unknown length of the corridor/30s) * year

So we have two unknowns and only one equation... So apparently they didn't give us the rest of information needed to check their calculation (and the clock was behind a glass, so I couldn't just measure the corridor on my own).

Well, I was just thinking, I can backwards calculate at least the length of the corridor and by-the-eye check if the result will fit my estimation how long the ball-corridor could roughly be...

Let's go back then:

s = v * t

2,500 miles = x/30s * year

converting units (metres, seconds, therefore m/s as a speed unit):

1 mile = 1.609344 kilometres => * 100 => 1,609.344 metres

2,500 miles = 1 mile * 2,500 = 1,609.344 metres * 2,500 = **4,023,360 m**

1 year = 365.25 days = 365.25 * 24 hours = 365.25 * 24 * 60 minutes = 365.25 * 24 * 60 * 60 seconds = **31557600** seconds in a average in every year

[I took it astronomically and made a mean for every year, because yes, that's why we have 1 additional day every 4 years => every year 1/4 - or 0.25 - of a day.

Otherwise there's 365 * 24 * 60 * 60 s = 31,536,000 s in a non-leap year and 366 * 24 * 60 * 60 s = 31,622,400 s in a leap year if not thinking about average.]

Again:

2,500 miles = x/30s * year**4,023,360 m** = x/30s * **31557600**

After two steps (dividing by 31557600 and multiplying by 30) we'll have:

(4,023,360/31557600) * 30 = x (x is in metres) = 3.82477755... m ≈ **3.82 m**

Hmm, was the length of the corridor for the ball counting minutes (half-minute one way) for that weird clock really over three and a half metres? I'd say it may have been a bit shorter, but thinking again about the size of the clock and the number of curves the ball must do when going from one end to the another one - yeah, it could be.

If you want to check it, you could either go to the British museum or at least google Congreve rolling ball clock and try to estimate it as I did.

So, this is what I think about when on holidays... I should really go for proper holidays now =)