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Mechanics 4 Coursework - page 4
Keywords: Modelling, Landing sequence, Aeroplane, Particle
By christi on 11/07/2009
Level: A Level (Year 13)
Page Number: 4 of 5 pages: 1 2 3 4 5the two models’ approximations of the point when t is 9, and that both curves proffer considerable underestimates, the runway length yielded by integration of said curves could not be trusted.
What is most apparent through carefully poring over the assumptions aforementioned is that the air resistance is not linearly proportional to the velocity. A more likely relationship is that it is proportional to v squared, and this is the axiom upon which model 2 rests.
Therefore the model is…
v' = -Av^2 when t T
where A, B and T are unknown constants.
Air resistance exerts a force that is proportional to the SQUARE of the velocity.
We want to estimate A, B and T.
Estimate of T
When t T we have
v' = -Av^2 - B
The data lets us estimate v'(10) as (1/2)(v(11) - v(9)) = -4.5, and so on:
t 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
v’(t) -4.5 -4.5 -4.0 -3.5 -3.5 -3.5 -3.5 -3.0 -3.0 -2.5 -2.5 -3.0 -2.5 -2.5 -2.5 -2.5
v (t) 50 46 41 38 34 31 27 24 21 18 16 13 10 8 5 3
Our estimate of B is the average of -v' - 0.00085847 v^2 taken over the values in the table:
(1/16)[(4.5 - 0.00085847(50)^2) + (4.5 - 0.00085847(46)^2) + ... + (2.5 - 0.00085847(3)^2)] = 2.51408
THE THIRD MODEL
v(0) = 1/0.010424 = 95.9325
v' = -0.00085847v^2 when t 9
Thus the final equation is exactly.
Hence the final equation is:
v' = -0.00085847v^2 when t 9
A plot of the final model may be seen below, which indicates that the brakes are applied around .
Finally, we compare the model with the data by solving the differential equations (the solution to the first DE gives an initial condition for the second) and plotting a graph. There is quite an accurate agreement.
As is apparent from the graph, this model is considerably more accurate than the first, and the closeness of the curve to the original data points suggests that the revised assumption for the air resistance is a sound one. Quantitatively how accurate 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 88.69 - 0.3502
2 82 82.41 - 0.5020
3 77 76.96 - 0.04600
4 72 72.19 - 0.2680
5 68 67.98 - 0.03162
6 64 64.23 - 0.3577
7 61 60.87 - 0.2108
8 58 57.85 - 0.2631
9 55 55.11 55.00 0.1994/0
10 50 - 50.10 0.2032
11 46 - 45.61 0.8531
12 41 - 41.45 1.104
13 38 - 37.58 1.096
14 34 - 33.96 0.1282
15 31 - 30.54 1.500
16 27 - 27.29 1.069
17 24 - 24.19 0.7962
18 21 - 21.22 1.046
19 18 - 18.35 1.970
20 16 - 15.58 2.633
21 13 - 12.88 0.9537
22 10 - 10.23 2.325
23 8 - 7.635 4.561
24 5 - 5.072 1.431
25 3 - 2.530 15.66
26 0 - 0 undefined
It follows from this table that the average errors to three significant figures are 0.223 and 2.20, over nine times less and over six times less than the average errors of the original model, even with the outlier of 15.66 included. Hence, having proven this model, it may be integrated to determine the length of the runway, L.
Alternatively the length of the runway could be calculated adding up the v values in the given table: 96 (distance travelled in 1st second) + 89 (distance travelled in 2nd second) +
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