### 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.

### Far Horizon

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

### Air Routes

Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.

# Alternative Record Book

##### Stage: 4 and 5 Challenge Level:

1. A human travels fastest in the men's 100m race. The record time is 9.58 s and if we substract an average reaction time 0.15 s then the average speed would be $38.2 km/h$. If we let a human use a bicycle then the event where they travel fastest would be track cycling. The speed is about $70 km/h$.

2. A human reacts fastest again in the men's 100m race. Reaction times are about 0.09 - 0.15 s.

3. An object travels fastest in a shooting event. The speed of a bullet travelling out of a rifle is about $3500 km/h$. Moreover, a ping pong ball can travel about $110 km/h$, a tennis ball travels about $150 km/h$, a javelin travels at $90 km/h$ and in badminton shuttlecocks travel at over $400 km/h$.

4. An object travels highest in a javelin throwing event. A javelin reaches a height of 50m or it could be a missed bullet which can reach a height of a few kilometres.

5. A human's head experiences about 60g acceleration in a boxing event.

6. A human expends the most calories in a cycling - road event. Cyclists are racing for about 7 hours and they approximate burn 9000 kcal.

7. Force. For example a hammer could be thrown for 80 meters. This means it has a speed of about 28 m/s just before being released. The hammer's mass is 7.257 kg and measures 1.215 m. Thus the force on the hammer $(7.257 \times 28^2)/(1.215) = 5 kN$. In a high jump event, the current record is $2.45 m$ which means that the jumper had a speed of  $6.9 m/s$. In the high jump, the acceleration off the ground takes about $0.15 - 0.2 s$. Thus, the force a jumper exterts on  the ground is $(80 \times 6.9)/0.16 = 3.5 kN$.

Stress is a measure of the average force per unit area. We calculated that the force on a hammer is about 5 kN and we know that the diameter of standard hammer throw wire is 4mm. Thus, the stress in wire is $5000/(1.25 \times 10^{-5}) = 400 MPa$. Moreover, there are high stresses in a pole vault pole about 1 GPa.

Strain is a measure of deformation. It is highest in a pole which is used in a pole vault event. It is about 2.5%. Read more.

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