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tua96426
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Homework Statement
Need to solve the differential equation which models the velocity of a sky diver.
Homework Equations
dv/dt = -9.8 + 0.0045Av^2 where A is cross sectional area (assume 0.75m^2)
v(0) = 0
The Attempt at a Solution
ok the problem is to find v(t) as a function time. as they teach us any ODE class, I started with separation of vars and on the left i had (dv/0.0045Av^2 - 9.8) and dt on the right,
then integrate both sides. The right hand side just becomes t + y (constant of integration) and the left needs to be simplified first with partial fractions
I said let c =sqrt(0.0045A) and b= sqrt (9.8)
then we have dv/(cv)^2 - b^2 which is now a difference of perfect squares ..
so partial fractions: A/(cv - b) + B(cv+b)
getting common denominator we get Acv + Ab + Bcv - Bb = 1
so all vars together: Ac + Bc = 0 >>> A+B = 0 therefore A = -B
then looking at the constant terms, I get b(A-B) = 1 >>> A = 1/b + B
so equation the two equations for A we get B = -1/2b and A = 1/2b
B = -1.565 and A = 1.565
so now we have Integral of (1.565/cv-b) + (-1.565/cv-b) dv
making a u substitution: u = cv +/- b >> du = cdv >>> dv = du/c
so we have (-1.565/c)ln|cv-b|+ (-1.565.c)ln|cv+b| = t + y
taking 1.565/c common on the left and multiplying by c/1.565 on both sides.. and getting all ln into one, I got
ln |(cv-b)/(cv+b)| = (c/1.565)(t+y)
let x = -c/1565
so doing more of mumbo jumbo, i get
v(t) = (b/c) ((1 + e^x(t+y))/(1-e^x(t+y)))
some how after applying the initial conditions, I got y = 0, but when I plug in 0 for t after getting the unique soln for v(t), I do not get 0 for velocity; rather I end up dividing by 0..
I researched this on wiki and they have the answer in tanh, which my prof said is possible. I was wondering how to get it in that form.. infact if I am able to express it in tanh it becomes easier for me to do the following problems (need to integrate and so on and so forth) .. any help on this would be gr8 peeps!
thanks
- Sudhi