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Mechanics 4 Coursework - page 2
Keywords: Modelling, Landing sequence, Aeroplane, Particle
By christi on 11/07/2009
Level: A Level (Year 13)
Page Number: 2 of 5 pages: 1 2 3 4 5equation impossible;
• In comparison to the air resistance, the pre-braking frictional force will be insignificantly small;
• It is extremely difficult to model weather accurately, and the extra forces entailed thus are unlikely to have a significant impact.
Therefore, the quantities used are:
T, time = independent variable
V, velocity = dependent variable
K , constant = parameter
Therefore, now I can formulate a mathematical model for this situation. The two variables I am using are v (velocity) and t (time). The air resistance is . The air resistance is proportional to the velocity which is between the aeroplane and the runway. Then the equation:
= -kv
comes out. (Weight is acting downwards on aeroplane so positive) upthrust acts upwards and so negative. Air resistance opposes motion of aeroplene so is negative, hence –kv)
4 HYPOTHESIS
I will use the above information to predict/hypothesize the model so that I can use that as a basis, to draw a conclusion from.
5 INVESTIGATION
THE FIRST MODEL
Substituting the data from result 1 into the equation for v:
When t = 0, v = 96, so
96 = Aeº = A
So in this case the constant of integration, A, is 96. The equation for v is now
V = 96e^-kt
When t = 1, v = 89
So 89 = 96e^-k
e^-k = 89/96 = 0.927083 …
k = -ln (0.927083 …)
k = 0.0757118…
k = 0.0757 (3sf)
so v = 96e^-0.0757t
=
=
=
= 120, 000(26) – (0)
= 31,200,000Ns
-Rt + c = 120 000v
Therefore the first model is -Rt + c = 120 000v. This does not reveal much about the nature of the forces acting on the plane and therefore must be further developed. To do this a graph of v against t must plotted in order to see more about the aeroplane and if possible when the brakes are applied. Below is a plot of v against t:
Since the data set does not indicate when the brakes are applied, and separate models are required for pre- and post-braking situations, it is necessary to determine between which points the brakes are applied. The graph above suggests that they are applied between 9 and 10 seconds, and between these two points the largest drop in the velocity occurs (5ms-1).
ln v 4.564 4.889 4.407 4.344 4.277 4.220 4.159 4.111 4.060 4.007 3.912 3.829 3.714 3.638
t 0 1 2 3 4 5 6 7 8 9 10 11 12 13
ln v 3.526 3.434 3.296 3.178 3.045 2.890 2.773 2.565 2.303 2.079 1.609 1.099 0
t 14 15 16 17 18 19 20 21 22 23 24 25 26
By further examining the graph of lnv against t, this suspicion is further confirmed with the noticeable splitting of the graph into two straight lines. This observation leads to the first model.
THE SECOND MODEL
Following
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