Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

Overarch 2

Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?

Cushion Ball

The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?

Dam Busters 1

Stage: 5 Challenge Level:

A bomber carrying a small, very heavy bomb is flying at speed $V$ at steady height $H$ above ground level. The pilot wants to strike directly at an enemy dam. The dam is on top of a rocky outcrop. The base of the dam wall is at a height $B$ above ground level and the top of the dam wall at a height $T$ above ground level.

This text is usually replaced by the Flash movie.

Given certain assumptions on the dynamics of the bomb, show that the bomb will strike the target directly if and only if $B< -\frac{g}{2}\frac{D^2}{V^2}+H< T\;.$

Now, a commander wishes to attack a real dam. The base of the dam is located at $100\mathrm{m}$ above ground level and the dam wall is $50\mathrm{m}$ high. The bomber travels at a top speed of $800\textrm{ km per hour}$ and must travel below $200\mathrm{m}$ to avoid radar detection. It must also release the bomb at least $1\mathrm{km}$ from the target to avoid the guns defending the dam. Given these constraints, can the bomber destroy the dam? What is the highest point above ground level that the bomb can actually strike? Use $g = 9.8 \mathrm{ms}^{-2}\;.$

Discussion / investigation : You made assumptions in the derivation of this result. For a real bomb there will be small corrections due to wind resistance and other factors. How would these affect your conclusions for the previous part?

Extension: Why not try the follow up problem Dam Busters 2 ?

NOTES AND BACKGROUND

Bombing dams and other key military targets was a real mathematical challenge during World War II. Bombs were dropped from planes and then simply fell under gravity, unlike the guided missiles of today. If a target were to be hit accurately, then the bomb would have to be released from the plane at very particular distances from the target whilst travelling at very specific speeds and heights. The range of defences of key targets were analysed carefully to try to find a way to deliver the bomb whilst minimising risk to the bomber crew. Since objects fall along parabolas under gravity the problems reduce to finding which parabolas join two points in space and then creating the conditions such that the bomb falls along that particular parabola.

The most famous bombing raid historically involved the creation of a spinning bomb which bounced on water, like a stone skimming across a lake. Some mathematical ideas surrounding this are described in the follow up problem Dam Busters 2.