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Mechanics 4 Coursework - page 3
Keywords: Modelling, Landing sequence, Aeroplane, Particle
By christi on 11/07/2009
Level: A Level (Year 13)
Page Number: 3 of 5 pages: 1 2 3 4 5this, a model can be constructed, assuming that the air resistance is proportional to the velocity. Diagrammatically, pre-braking, where A is in the negative direction:
A
W = mg = 120000g
R
where R is the air resistance. By Newton’s second law, that links the motion of the aeroplane to the forces acting on it.
The resultant downward force is 120000g – A.
Applying Newton’s second law, F=ma at any instant gives:
120000g – A = 120000
Dividing both sides by 120000 gives:
g – =
The modeling assumption that A is proportional to v, gives A = -kv and so
= g - = g +
Since k can and shall be reused throughout the post-braking model in addition to the pre-braking model, it is imperative that the constant be as accurate as possible. In order to maximise accuracy, each value of t in the range shall be averaged; is excluded as it does not yield a value for the constant.
Diagrammatically, where A+B is in the negative direction:
A+B
W = mg = 120000g
R
Assuming again that the air resistance is proportional to the velocity, by Newton’s second law [that :
Therefore because averaging and evaluating the constants exactly would be impractical, and the compounding of errors inherent in obtaining an average numerically, a compromise has been sought in the form of using the two extreme values of t in the range as an intermediate. Since the most significant constant, k, has already been averaged, and two well-spaced values have been chosen, the model will not suffer irreparably.
Quantitatively how much the curves deviate from the original data is tabulated below, to four significant figures (where appropriate):
t V Pre-braking approximation Post-braking approximation Percentage error
0 96 96 - 0
1 89 89.55 - 0.6157
2 82 83.53 - 1.8652
3 77 77.92 - 1.189
4 72 72.68 - 0.9430
5 68 67.79 - 0.3026
6 64 63.24 - 1.191
7 61 58.99 - 3.300
8 58 55.02 - 5.132
9 55 51.33 55 6.682 (1)/0 (2)
10 50 - 49.67 0.6594
11 46 - 44.70 2.829
12 41 - 40.06 2.289
13 38 - 35.74 5.958
14 34 - 31.70 6.762
15 31 - 27.94 9.880
16 27 - 24.43 9.532
17 24 - 21.15 11.87
18 21 - 18.10 13.83
19 18 - 15.25 15.29
20 16 - 12.59 21.32
21 13 - 10.11 22.23
22 10 - 7.797 22.03
23 8 - 5.640 29.50
24 5 - 3.628 27.44
25 3 - 1.751 41.64
26 0 - 2.340×10-13 Undefined
As is evident from the rapid increase of the percentage errors, averaging all of the constants in the pre-braking model made a significant difference. By neglecting to do this for the post-braking model, it has resulted in a curve which perpetually underestimates the values of the empirical data, something which is decidedly unsafe for a runway length, and thus the integral will have to be scaled up accordingly. Given that the average percentage errors to three significant figures are 2.12% and 14.3% for the pre-braking and post-braking models respectively, the final integrals could be scaled up by this much to partially mitigate accumulated error; however, given that there is almost a 7% difference between
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