### 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

### Why do this problem?

This problem gives a different approach to the standard problem of motion of a projectile. It also involves ideas from inequalities. The problem can be used to encourage estimation in a difficult situation.

### Possible approach

This problem works best as an individual task for the first part.

For the second part, students could be asked to try to decide whether the bomb will strike the dam, before undertaking a detailed calculation based on the figures alone. You could help this process by asking how long (roughly) the bomb will take to hit the dam and how long (roughly) the bomb will take to hit the ground. Given these estimates, will the bomb have dropped enough in time to hit the dam?

### Key questions

• How will we model the motion of the bomb?
• Do you think that the bomb will hit the dam?
• How long will the bomb take to travel far enough to hit the dam if released at a distance of 1km?
• How will releasing from a lower height/further distance/faster speed affect the point of impact?

### Possible extension

The natural follow up to this question is Dam Busters 2

### Possible support

Ask students to solve the equation for the specific case of the bomb released at $1$km and from $200$m at $800$ km per hour.