A parachutist weighing m=75kg jumps from an airplane at an altitude of h meteres with initial vertical velocity v0=0. Let v(t) be his instantaneous vertical velocity at time t with the downwards direction taken as positive. Suppose that he deploys the parachute immediately after jumping from the airplane. The air resistance is assumed to be r(t)=15v^2(t). Let g=9.8ms^-2 be the acceleration due to gravity. We neglect side wind. The equation of motion is
mdv/dt=F
a) Show that the instantaneous vertical velocity satisfies dv(t)/dt + v^2(t)/5 = 9.8
b) solve the initial value problem and determine the instantaneous vertical velocity v(t)
c) determine the impact velocity before landing. Express it in terms of h.
I CANT GET B AND C MAN :'( been trying whole of last night and this morning.
for c it asks for impact velocity in terms of h
is my answer correct
I got
v(h)=sqrt(g/k) tanh(arccos(e^(h.sqrt(k/g)))
can I rewrite the arccosh(x) as ln(x+sqrt(x^2-1))?
next question if anyone can provide guidance would b appreciated
Consider the differential equation for y(t) (t>=0)
dy/dt=sech(y)-2e/(e^2+1)
a) find the equilibrium solutions (is it 1 and -1?)
b)draw a phase plot of dy/dt versus y and label all intercepts (upside down parabola style at -1/1 intercepts right?)
c) Determine whether the equilibrium solutions are stable or not (is -1 unstable, and 1 stable?)
d) draw rough sketches of y(t) against t for several initial values y(0). Choose these initial values so they are spread across the regions defined by the equilibrium solutions and make sure that you illustrate all of the main shapes of population versus time graphs that can occur. You should draw a minimum of 4 such curves (NO IDEA PLS HALP)
a) Yes.
b) Pretty much.
c) Correct!
d) Have a play around with this: https://www.geogebra.org/m/W7dAdgqc (type in sech(y) - sech(1) for the f'(x,y) in the top left).
Basically you need to sketch the direction field and draw some solution curves passing through various different points on the y-axis (corresponding to different values of y(0)). You should be able to see that depending on where you choose your y(0) value (y-intercept) in relation to the two bands of stable solutions, the solution curve that results will behave differently. If you have y(0) > 0, the solution will approach 1 as t -> oo. If we have -1 < y(0) < 1, the solution will approach 1 as t -> oo. If we have y(0) < -1, the solution will go to -oo as t -> oo.
In that applet I linked, you can tick all the boxes for Solution A-D in the bottom left to get various solution curves, and you can drag the points they are forced to go through to different places to see how the resulting solution curve changes.
how did you identify y for part c?
Just find the roots of the RHS function of the DE y' = f(y).
You can see here for more information and examples: http://tutorial.math.lamar.edu/Class...Solutions.aspx .
Last edited by InteGrand; 30 Apr 2017 at 12:18 PM.
InteGrand if you can help with this would be amazing.
1. Find the general solution to the differential equation y''+2y'+3y=1+t^2+e^-t
is y(t)=Ae^-t.cos(sqrt2)t+Be^-t.sin(sqrt2)t+(e^-t)/2+(t^2)/3-(4t/9)+17/27?
and 2.Consider an electric circuit
L
-------<<------
i i
[V] i
i i
i i
-------II--------
C
With an inductance of L=0.5 Henry and a capacitor with a capacitance of C=0.1 farad connected in series with a voltage source of V(t)=50sin(wt) Volts. Initially the charge on the capacitor is 2 Coulombs and there is no current in the circuit.
a) Show that the differential equation satisfied by the charge q(t) in this circuit is q''+20q=100sin(wt)
b) Determine the value(s) Wres for which resonance occurs
c) Solve the initial value problem to find the charge qNR(t) as a function of time in the case where there is NO resonance.
d) Solve the initial value problem to find the charge qR(t) as a function of time for ONE of the values w(res) where there IS resonance (You can choose which wres value to consider.
Question 2 idk :'(
For Q1) you basically find a particular solution and add it on to the solution to the homogenous equation to get the overall solution. I don't want to do the calculations now but I assume you know the steps.
And for Q2), you should probably review your resonance theory (and did you manage to set up the ODE? Did you try Kirchhoff's Laws?). Here's some info and examples on resonance: https://math.libretexts.org/TextMaps..._and_Resonance .
so if sin(wt) is equal to the general solution values of sin and cos?
characteristic equation results in 20i,-20i so y(t)=Acos20t+Bsin20t, if times by t will be equal with sinwt so
20 is it? later questions say "pick a value where resonance occurs" isnt there only the one?
Yeah there's only one positive value, but you could also use -20 and that'd still give resonance. Usually omega is thought of as positive I guess. Using a negative omega here just makes the RHS of your ODE become negative sine, since sine is odd. This means your ''particular solution'' becomes negative of the old one.
Last edited by InteGrand; 9 May 2017 at 9:32 PM.
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