Four vehicles travelled on a road with constant velocities. The car
overtook the scooter at 12 o'clock, then met the bike at 14.00 and
the motorcycle at 16.00. The motorcycle met the scooter at 17.00
then it overtook the bike at 18.00. At what time did the bike and
the scooter meet?
Two trains set off at the same time from each end of a single
straight railway line. A very fast bee starts off in front of the
first train and flies continuously back and forth between the two
trains. How far does Sidney fly before he is squashed between the
Brian swims at twice the speed that a river is flowing, downstream
from one moored boat to another and back again, taking 12 minutes
altogether. How long would it have taken him in still water?
This superb solution came from James of
Hethersett High School, Norfolk, well done James.
On the video the wheels appear to be going backwards as the
reflector is not making a full turn between shots, but just
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
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
Note: This is 6.2333/6.2832 = 0.992
revolutions per half second.
Here James worked out how fast the cyclist
would have to go for exactly one revolution of the wheels each half
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:
so, to make the wheels look stationary, the cyclist has to cycle
at 8.5680 miles per hour (to 4 decimal places).
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
First I had to work out the circumference of the wheel (60p
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:
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
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.
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.