Hello. I've been doing my differential equations coursework recently and I am so close to finishing it but I have one last bit I'm stuck on.
The coursework aim is to model 2 equations, representing the velocity of an aeroplane that has just landed (at t=0). All we are given is a table of particular values for the velocity of the plane over 26 seconds (at 1s intervals), and the mass of the plane (120 000 kg).
Initially, the plane is slowed by just air resistance, then by air resistance and a CONSTANT breaking force.. so before breaks are on:
v' = -kv2 (air resistance is proportional to velocity)
Solving this gives me 1/v = kt - c
at t=0, v=96 => c = -1/96
I used parameters v=55,t=9 (given to me in the table) to find the value of k (these gave the smallest error across all the particular values I am supposed to be modelling)
I have to get it to the nearest integer (that's how the data is given to me), and with these parameters I managed to model all of the data correctly before t=9 (This is when I worked out that the breaks were turned on). My value for k was 41/47520 (=8.6279461x10-4)
My problem is after t=9.
I decided that since it is a constant breaking force, the deceleration associated with that constant braking force will also be constant. So...
v' = -kv - B
rearranging and seperating variable:
INTEGRAL OF(1/kv2+B)dv = INTEGRAL OF(1)dt
Obviously the right hand side is easy to integrate, however the left hand side is the problem.
I tried using the fact that INTEGRAL OF(1/a2+x2)dx = (1/a)arctan(x/a) + c
This gave me (although I may have made a mistake somewhere)...
(kB)-1/2arctan(v(k/B)1/2) + c = -t
when v=0, t=26 => c = -26
so... finally I have
(kB)-1/2arctan(v(k/B)1/2) - 26 = -t
(remember, I know the value of k already, B is the only unknown constant).
How on earth do I work out what B is?! (without use of trial and improvement/trial and error)
Thank you for your time!
P.S... Here is the table of values I was given:
t - v
0 - 96
1 - 89
2 - 82
3 - 77
4 - 72
5 - 68
6 - 64
7 - 61
8 - 58
9 - 55
10 - 50
11 - 46
12 - 41
13 - 38
14 - 34
15 - 31
16 - 27
17 - 24
18 - 21
19 - 18
20 - 16
21 - 13
22 - 10
23 - 8
24 - 5
25 - 3
26 - 0
The coursework aim is to model 2 equations, representing the velocity of an aeroplane that has just landed (at t=0). All we are given is a table of particular values for the velocity of the plane over 26 seconds (at 1s intervals), and the mass of the plane (120 000 kg).
Initially, the plane is slowed by just air resistance, then by air resistance and a CONSTANT breaking force.. so before breaks are on:
v' = -kv2 (air resistance is proportional to velocity)
Solving this gives me 1/v = kt - c
at t=0, v=96 => c = -1/96
I used parameters v=55,t=9 (given to me in the table) to find the value of k (these gave the smallest error across all the particular values I am supposed to be modelling)
I have to get it to the nearest integer (that's how the data is given to me), and with these parameters I managed to model all of the data correctly before t=9 (This is when I worked out that the breaks were turned on). My value for k was 41/47520 (=8.6279461x10-4)
My problem is after t=9.
I decided that since it is a constant breaking force, the deceleration associated with that constant braking force will also be constant. So...
v' = -kv - B
rearranging and seperating variable:
INTEGRAL OF(1/kv2+B)dv = INTEGRAL OF(1)dt
Obviously the right hand side is easy to integrate, however the left hand side is the problem.
I tried using the fact that INTEGRAL OF(1/a2+x2)dx = (1/a)arctan(x/a) + c
This gave me (although I may have made a mistake somewhere)...
(kB)-1/2arctan(v(k/B)1/2) + c = -t
when v=0, t=26 => c = -26
so... finally I have
(kB)-1/2arctan(v(k/B)1/2) - 26 = -t
(remember, I know the value of k already, B is the only unknown constant).
How on earth do I work out what B is?! (without use of trial and improvement/trial and error)
Thank you for your time!
P.S... Here is the table of values I was given:
t - v
0 - 96
1 - 89
2 - 82
3 - 77
4 - 72
5 - 68
6 - 64
7 - 61
8 - 58
9 - 55
10 - 50
11 - 46
12 - 41
13 - 38
14 - 34
15 - 31
16 - 27
17 - 24
18 - 21
19 - 18
20 - 16
21 - 13
22 - 10
23 - 8
24 - 5
25 - 3
26 - 0