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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?

Ramping it Up

Age 16 to 18 Challenge Level:
You might think about the graphs in terms of gradients.

You could consider replacing the vertical lines by very steep gradient lines.

To relate the problem to acceleration, recall that the acceleration is the second derivative of the position.

Don't forget that you are not allowed to have 'infinite' values on a graph.

This problem introduces the mathematics behind 'step functions' and 'Dirac delta' functions.