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# Powerfully Fast

NOTES AND BACKGROUND

There are various parts to this problem, which all use realistic numbers. Conversion of units will be required at various points. Be sure to apply common-sense checks at each point to be sure that you have not made numerical errors -- many engineering disasters have occurred because of problems with conversion of units!

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Age 16 to 18

Challenge Level

The power-to-weight ratio (or specific power) of an engine is defined to be the maximum power an engine can produce divided by the mass of the engine.

A certain jet engine has a mass of $4000\textrm{ kg}$ and a power to weight ratio of $12000\textrm{ Js}^{-1}\textrm{ kg}^{-1}$. An aeroplane of unloaded weight $140$ tonnes contains two of these engines.

What is the overall power to weight ratio of the plane?

Neglecting friction, how long would it take for the plane to accelerate from $0$ to $60$ miles per hour? How long to accelerate from $120$ to $180$ or $180$ to $240$?

A spaceship is made with the same weight and power as the aeroplane. Its boosters can burn for 8 hours without refuelling and contain $120000$ litres of fuel, of density $0.81715\textrm{ kg l}^{-1}$. If the spaceship is launched from in space (so that we can ignore friction and gravity), what is the maximum velocity that the spaceship can achieve? (Note: the variation of mass in this part is
significant. To start, make an approximation assuming constant mass. You might then consider the variable mass problem, which will involve harder calculus).

Watt found by experiment in 1782 that a 'brewery horse' (presumably a large draft horse) was able to produce $32400$-foot-pounds per minute. Estimate the power to weight ratio of a horse. How many such horses would be needed to power the aeroplane above? Assuming constant production of power, how long would it take such a horse to accelerate from $0$ to $60$ miles per hour?

Investigation: Explore the power to weight ratios of various interesting objects such as the spaceshuttle, Concorde, Ferraris, sprinters, cheetahs, milkfloats and tortoises.Which produce the best acceleration over various ranges of velocities?

NOTES AND BACKGROUND

There are various parts to this problem, which all use realistic numbers. Conversion of units will be required at various points. Be sure to apply common-sense checks at each point to be sure that you have not made numerical errors -- many engineering disasters have occurred because of problems with conversion of units!

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?

Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.

Given the graph of a supply network and the maximum capacity for flow in each section find the maximum flow across the network.