Bike forwards, Wheels backwards

[steveh]

Evening all

I was Googling for something entirely different and came across a puzzle / illusion, after an hour of not being able to find the answer and having used half a rainforest to aimlessly doodle numbers etc. I resorted to looking at the answers page. Blow me down with a feather a schoolboy answered it!! I am sure therefore that this is no tester for you geniuses out there but I guess I should throw down the gaunlet for the quickest answer

A security camera, which takes one picture each half a second, films a cyclist going by at 8.5 miles per hour on a bike which has wheels of diameter 24 inches. There is a reflector fixed to the spokes of each wheel of the bike. Explain why it is that, in the film, the cyclist appears to go forward while the wheels appear to go backwards. If the cyclist goes a little faster the wheels appear to be stationary. At what speed does this happen?

You will need to know that 1 mile = 1760 yards, 1 yard = 3 feet and 1 foot = 12 inches to do this question.

Can you also change the numbers (not necessarily by direct conversion) to pose a similar problem in metric units?

The stop clock has started at 16:41 GMT

[HansV]

Spoiler

A wheel with a diameter of 24 inches has a circumference of pi*24 inches (where pi=3.141592...), or ~ 75.4 inches.
In ine hour, the cyclist travels 538560 inches, so in half a second, he or she travels 538560 / 7200 ~ 74.8 inches.
This is slightly less than one revolution of the wheel, so the reflector will appear to go slowly backwards.

For the reflector to appear stationary, the cyclist must travel 75.4 inches per second (or a multiple thereof), which corresponds to ~ 8.568 miles per hour.

[HansV]

A roughly equivalent problem in metric units would be: the cyclist travels at 13.5 kilometers per hour, and the wheel has a diameter of 60 cm.

[steveh]

Oh Boy, thats good

Ask a teacher a kids questions and what did I expect :-)

[Bigaldoc]

And for those of us not so mathematically inclined, where is this "puzzle" please?

[steveh]

And for those of us not so mathematically inclined, where is this "puzzle" please?

You mean like me Al?

This is the site I found it on

http://nrich.maths.org/628" onclick="window.open(this.href);return false;

[Bigaldoc]

Just for the heck of it, since I didn't yet know the site, I went Googling last night to see if I could find a video of such a phenomenon. Here's one that I thought was pretty cool although it isn't exactly like your puzzle:

YouTube - MonkeyLectric Video Pro bike wheel light display

[steveh]

Wow

Thats really cool, better post it in Scuttlebut for Bowlie, I off to get my old trike out of the loft

[Sundog]

So when are the filmmakers of car commercials going to have both the car and the wheels going in the same direction?

[HansV]

So when are the filmmakers of car commercials going to have both the car and the wheels going in the same direction?

That's not really necessary - most people have become used to the effect.

[Sundog]

Mebby so, but it always makes me laugh. Following the SteveH logic, it seems like they ought to film the cars going faster?

[HansV]

You could perform the same calculations for a car; the actual diameter of the wheel depends on
- The diameter of the wheel rim.
- The tire.
- The load on the tire.

A reasonable figure for a 'normal' car would be a diameter of about 30". The wheel would turn a whole number of revolutions in a half second if the speed is a multiple of ~2.68 mph, so for example at 12 * 2.68 ~ 32.16 mph. At that speed the wheel would appear to be stationary. If the car goes slightly faster than that, the wheel would appear to rotate forwards; if the speed is slightly less, the wheel would appear to rotate backwards.

[Sundog]

OK, now all we need is the shutter speed for a commercial film camera (or video camera scan rate?). Anyone have Hollywood connections to answer this?

[HansV]

See Film frame on Wikipedia. 24 frames/second appears to be the most common. The example in my previous reply corresponds to 24 revolutions/second at 32.16 mph (I chose that multiple on purpose).

[Sundog]

Excellent! Very smart choice! (So why don't they drive the cars at 35 MPH during commercial-making? )

[HansV]

35 mph is probably too staid...

[Sundog]

Oh, I don't know. How fast is going? It seems to have blown all his/her/its hair off.

[HansV]

For some reason, never seems to get very far...

[Sundog]

Neither do the cars in the commercials... Maybe if I had a realllly biiigggg screeeennn?

[rettingr]

Would the same formula work on this bike?