Illusion
Problem
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?
Student Solutions
This superb solution came from James of Hethersett High School, Norfolk, well done James.
Method 1
On the video the wheels appear to be going backwards as the reflector is not making a full turn between shots, but just less.
To work out this question I first worked out how many feet there are in a mile.
1 mile = 1760 yards and 1 yard = 3 feet so there are 5280 feet in a mile.
As the cyclist is travelling at 8.5 miles per hour I multiplied 5280 by 8.5 and found that travels 44,880 feet in 1 hour. Then I calculated how many feet he travels in 1 second then in half a second.
44,880/60 = 748 so he travels 748 feet in 1 minute.
748/60 gives the speed in feet per second and, dividing by 2, I worked out that he travels 6.2333? (6.23 recurring) feet every half second. Now I have to work out the circumference of the wheel.
24 inches = 2 feet
The circumference is 2p = 6.2832 ft (to 4 decimal places).
6.2832 - 6.2333 = 0.0499.
The wheel looks as if it is going backwards as each time the video camera takes a picture it is 'stopped' just short of a full rotation.
Note: This is 6.2333/6.2832 = 0.992 revolutions per half second.
Method 2
Here James worked out how fast the cyclist would have to go for exactly one revolution of the wheels each half second.
The circumference of the wheel is 2p so this speed is 2p feet per half second or 4p
feet per second and converting this to miles per hour gives:
| 4p x 60 x 60 | ||
|
|
= | 8.5679996 |
| 5230 |
so, to make the wheels look stationary, the cyclist has to cycle at 8.5680 miles per hour (to 4 decimal places).
Method 3
To make the question work in metric units James said that the cyclist was travelling at 8.5 km per hour (leisurely pace) and the diameter of the wheel was 60 cm (close to 24 inches).
First I had to work out the circumference of the wheel (60p centimetres).
As the cyclist is travelling at 8.5 km per hour I have to multiply this by 1000 to work out how many metres he travels per hour, then how many metres he travels in one minute, then in one second then in half a second which gives:
| 8.5 x 1000 | ||
|
|
= | 1.18055555... |
| 60 x 60 x 2 |
So he travels 1.1806 metres every half second (to 4 decimal places). The wheel circumference is 188.50 cm = 1.8850 m (to 4 decimal places).
Note: This is 1.1806/1.8850 = 0.626 revolutions per half second. To make a viewer of the film think the wheels are going backwards this needs to be closer to one revolution per half second.
Method 4
Here James worked out the speed when the 60 cm diameter wheels complete exactly one revolution every half second making it appear on the video film that the wheels are stationary. This calculation gives 13.5717 kilometres per hour (to 4 decimal places). So if he cycled at 13.5 km per hour it would again appear as if the wheels were going backwards.