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Mechanics 4 Coursework - page 1
Keywords: Modelling, Landing sequence, Aeroplane, Particle
By christi on 11/07/2009
Level: A Level (Year 13)
Page Number: 1 of 5 pages: 1 2 3 4 51 INTRODUCTION
Differential Equations are normally used to model rates of change.
For example, an electron might travel through a container. It is affected by gravity, by various charges, and by magnetic fields. Gravitational and electrical affects depend on its position, and magnetic affects depend on its velocity. All the affects feed back into its acceleration, which immediately influences its velocity and position. This complicated feedback loop gives us an equation involving a(t), v(t), p(t), and t. Because the equation involves lots of derivatives, we call it a differential equation. To solve the equation is to find a formula for p(t) in terms of t, which is an exact prediction of the electron's future path.
In the example above, the differential equation feels like the exact description of the process itself. Often it comes directly from a general law of physics. Solving the equation tells the implications of the laws, and describes the expected consequences of the rules of the particular system.
In this investigation, the differential equation is modelling the rate of change in the frictional force when an aeroplane lands.
2 AIM
For this coursework, I have been tasked with modelling the landing sequence of an aeroplane, and determining a suitable runway length so as to ensure a successful landing. I will be investigating the differential equation that suitably models the nature of the forces acting on the plane during a period of 26 seconds.
The following tables depict empirical values of the aeroplanes speed, vms-1, t seconds after touchdown:
v 96 89 82 77 72 68 64 61 58 55 50 46 41 38
t 0 1 2 3 4 5 6 7 8 9 10 11 12 13
v 34 31 27 24 21 18 16 13 10 8 5 3 0
t 14 15 16 17 18 19 20 21 22 23 24 25 26
3 SIMPLIFYING THE SITUATION AND SETTING UP THE MODEL
Assuming that, in order of descending importance:
The aeroplane obeys a particle model; (treating aeroplane as particle)
There is no driving force;
The runway is a flat, uniform plane;
The only forces present in the direction of travel are air resistance and the brakes of which the braking force is constant;
The frictional force is discounted and ignored before the brakes are applied
The weather is moderate and temperate.
Taking into account that:
If the aeroplane were not to be modelled as a particle, moments would need to be considered, and for a normal, successful landing which in these circumstances will not be necessary;
A driving force would affect the parameters, and not the differential equation type;
A runway not conforming to the above characteristics would cause unpredictable and unreliable fluctuations in velocity;
Not restricting the forces may introduce additional independent variables, and so makes the solution of the differential
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