Ski-jumping
If you watched any of the TV transmissions from the Winter Olympics you must surely have impressed by the ski jumping.
As you probably recall, the lone British competitor 'Eddie the Eagle' came last in competition but won a great publicity victory for his courage.
What struck me as rather surprising was the margin between Eddie's performance and that of Finnish Olympic Medallist:
Eddie managed to jump only about half as far as the Finn. To the separator it looks as though in ski-jumping what the competitor does is simply let to go from the top of the ramp: the rest seems to be due of gravity. What did the Finn do differently from the Eddie that enabled him to jump so much further?
Devise a mathematical model of ski-jumping, which will enable you to find the length of a jump in terms of the jumper's speed as he leaves the ramp, and use your model to advise Eddie on how to improve his standard.
Firebreaks in forest
When large areas of land are panted as a forest, consideration has to be given to protecting the forest from devastation by fire. Once a fire starts it is normally only possible to stop it by creating firebreaks to contain the fire. Rather than create firebreaks when a fire breaks out it is now accepted that firebreaks should be incorporated from the start in the form of wide avenues devoid of trees. Then if a fire does break out, it will be confined to within the area between the firebreaks. By having many avenues to act as firebreaks, any fire will be contained to a smaller area and so the loss will be less. However in having too many avenues within the forest, the amount of trees that will eventually be harvested will be reduced.
Develop a mathematical model, which will predict the optimum spacing of firebreaks within a forest.
Cooling and polystyrene
The invention of polystyrene has brought many benefits. One of them is its use in the construction of containers such as picnic boxes for carrying quantities of food and drink and keeping it cool and fresh. It is the good insulation properties of polystyrene, combined with its lightness and rigidity, which make it such an excellent material for this purpose.
Imagine that you are employed as a designer in a plastic factory that plans to start manufacturing picnic boxes. Devise a mathematical model of the gradual warming of food and drink taken from a refrigerator and placed in a box of suitable size insulated with polystyrene; decide on a suitable performance target for a picnic box, and use your model to estimate the thickness of polystyrene insulation required to meet this target.
The problem is to work out how the yellow lines painted across the carriageway of a road on the approach to a roundabout should be spaced, in order to encourage drivers to decelerate so as to come to rest on arrival at the roundabout. The distance between successive lines should decrease as one approaches the roundabout, in such a way that someone driving with suitable constant deceleration will see the lines come past at constant rate. (That is, someone driving with suitable constant deceleration will take the same time to pass between each pair of successive lines).
Pollution of Great Lakes
Industrialised nations are beginning to face the problems to water pollution. Once pollution if river is stopped, the river will clean itself fairly rapidly if the pollution has not caused extreme damage. Lakes present a problem, because a polluted lake contains a considerable amount of water, which must somehow be cleaned. The only presently feasible method is to rely on natural processes. How long does this take? In particular, ho long would it take to clean up the Great Lakes?
Pollution affects a lake in many complex ways. Some of compounds such ad DDT enters biological systems and move up the food chain. Since DDT is very soluble on fat, it concentrates in the fatty tissue of higher predators and is hard to remove from the biosphere. Some pollutants more rather freely in and out of the food chain. The behaviour of phosphorus lies somewhere between these two extremes. (In one sense, phosphorus is not a pollutant, since it occurs naturally; however, excessive amounts can trigger algae blooms, and it is then considered a pollutant.) Still other pollutants, like oil spills, may be only slightly involved in the food chain. Extensive pollution can cause irreversible damage and even "kill" a lake.
The main cleanup mechanism is the relatively straightforward natural process if gradually replacing the water in the lake. In addition, other processes such as sedimentation and decay may be important.
Number of years required to drain the lakes if outflow is unchanged and inflow stops.
Superior | 189
|
Michigan | 30,8
|
Huron | ?
|
Erie | 2,6
|
Ontario | 7,8
|
Handcapping weightlifters
Weightlifting is a minority sport, but is a sport, which can be understood readily by the layman.
There are nine official bodyweight classes and there are two principal lifts namely the snatch and the jerk. In the snatch, the weight is pulled for the floor to a locked arm overhead position in a single move although the lifter is allowed to move or squat under the weight as it is being lifted, whereas for the jerk, two movements are possible, the first to the chest and the second to the overhead position.
It is clear that as the bodyweight increases so does the lift but what is not clear is the functional form of this increase. The problem is to design some form of handicapping, which compensates for the various bodyweight.
One of the major motorcar manufacturers has been showing an intriguing TV advertisement for one of its models recently. This car is fitted with a driver's airbag, and the purpose of the advertisement is to draw the viewer's attention to the benefits of having a airbag. In the advertisement, a car (of the relevant model) is driven very slowly off the flat roof a tall building. It is seen falling more-or-less vertically downwards towards what appears to be a piece of white material on the ground where it would land. Just before the car is about to smash into the ground, however, the piece of material begins to move: it turns out to be a giant airbag, which inflates fully just in time to arrest the car, which lands on top of the bag at the exact moment when it is fully inflated. The bag then gently subsides, and the car is driven off without having suffered the slightest damage.
Supposing that there was on electronic wizardry involved in the production of the advertisement, and that the events it shows actually happened as they appear to have happened, one cannot help wondering how the film-makers knew when to start blowing up the airbag. It would have been important to get the timing just right. If the inflation of the airbag had been left too late, then the pressure of gas in the bag wouldn't have been sufficient to stop the car hitting the ground. On the other hand, the airbag begins to deflate automatically as soon as it has reached its fullest extent, so as to land the car gently (it behaves just like airbags inside cars in this respect though they deflate in order not to obscure the windscreen any longer than is necessary). So if inflation had begun too early, the bag would also have been too soft to arrest the car effectively.
You are invited to use your mathematical modelling skills to solve problem of when to start inflating the airbag
Buying a car
It was recently started in a reputable economics journal that the decision to buy a car is second only to that of buying a house. Once owning a car was a luxury but now it is argued that it is almost a necessity. Unfortunately, as with houses, there are a bewildering number of options open to a prospective car buyer. Fantastic offers on used cars, cheap hire purchase, cash discounts even free holidays are available at most garages. But cars do not seem to last as long as they used to and the resale value of new car drops alarmingly after purchase. Buying an old car has the advantage that it does not depreciate at the same rate, also various other costs are cheaper, however, it is not as an attractive proposition as a new car. Could you help a typical car buyer in his dilemma.
"Is it best to buy an older car and run it till it completely breaks down or buy a new car and change it every two years"?
Double-glazing
Recent surveys have shown that many more householders are installing double-glazing although it is commonly held, especially in the north, that bedrooms should be fresh and aired i.e. no need for double-glazing upstairs.
Various firms claim that double-glazing is a worthwhile investment and the customer saves his initial outlay within a certain number of years.
Use a model to investigate the cost effectiveness of double glazing. What advice would you give to a newly married couple who have bought their first single-glazed home?
Bungee Jumping
This an increasingly popular sport. The skill is to just touch the surface if the water. The height if the tower is known but common sense dictates that the weight of a person must affect the type of bungee used.
If you were managing the bungee jumps, how would you account for this?
THE TREATMENT OF POLLUTED WATER
A CASE STUDY
Statement of the Problem
Polluted water is discharged from the steelworks at Throup and the
management have been ordered by the authorities to install an effluent
system to 'clean up' the pollution before the water enters the river Semi
Dee. In particular this system must ensure that the concentration of
pollutant is less than 5x10-4 g/m3 at all times. This could be a problem
since the plant discharge concentration varies between 10-3 and
10-2 g/m3.
Biological treatment systems are generally considered to be the most
reliable at dealing with the steelworks' pollutant and one company ( Poogone
Inc.) have submitted a design utilising a single well-mixed tank with a
volume of 10000 m3 coping with a flow rate through the system of 10
m3/hr.
Some concern was expressed by the Steelworks management at the tank size
recommended by Poogone. This concern generated two questions:
- Will the proposed scheme always work?
- Could a more reliable system using two much smaller tanks be designed?
Write a report to provide some answers to these questions, and to make
recommendations on the system to be used and its dimensions.