Lab 3 Skydiver

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This lab involves modelling a skydiver falling through the air.  Initially, air resistance is very small, but upon deployment of a parachute, air drag suddenly increases to a large value.  The mass of the jumper is 60kg, acceleration due to gravity is 9.81 m/(s^2) , the jumper starts from a height of 1000m and opens the parachute at about 200m above the ground.  The jumper feels a drag force F=-k v |v|, where k changes to a high value after the parachute is opened.

Start by removing all variables from the Global context.

To model the action of the parachute we will use the hyperbolic tangent function.  Plot tanh(x-5) from x=0 to 10.  Don't worry about making the graph look nice, but make sure the range of the graph goes from -1.2 to 1.2 using the PlotRange option of Plot (use Help if you need to).

As you can see from the graph, to a good approximation the function goes from one constant value to another.  Now plot the function 6 - 2 tanh( (x-5)/0.1 ) from x=0 to 10.

Notice how we have tranformed the function so that it changes more abruptly, has a wider range and different end values.  Let's now choose appropriate parameters to model our system, and store them in variables
m=60  - mass of jumber in kg
g=9.81  - acceleration due to gravity
h=1000   - initial height in m
y0=200  -  height at which the jumper opens the parachute
kd=0.48  -  drag coefficient without parachute in kg /m
kp=30.24 -  drag coefficient with parachute fully deployed
d= 20  -  roughly the distance (m) the jumper falls during the parachute opening

Define the function k(y) = kp - (kp-kd)/2 (1+tanh( 2 (y-y0)/d  )).  k(y) is our methematical model for the drag coefficient as a function of the jumper's height above the ground.

Plot the function k(y), with y ranging from 0 to 1000.

Check your values for k(0) and k(1000).  They should be 30.24 and 0.48 respectively.

Now that we've modelled the skydiver problem with a height-varying drag coeffiecient, let's find the height and velocity as functions of time using NDSolve, assuming initial velocity of zero. 

The equation that we want to solve is y'' = -g - k(y)/m y' | y' |, along with appropriate initial conditions.  Fill in the blanks.

Y[t_]=y[t] BLANK NDSolve[{y''[t]== BLANK, y[0]== BLANK, BLANK==0},BLANK,{t,0,100}]//First

Plot the height as a function of time (include axes labels with units).

Use FindRoot to find the total time the skydiver spends in the air.

Plot the velocity of the skydiver as a function of time (include axes labels with units).

What is the approximate terminal speed of the skydiver before the parachute is deployed?  What about after?

Plot the acceleration of ths skydiver (include axes labels with units).

What is the largest acceleration felt by the skydiver during his descent?  Use the function FindMaximum.   Express your answer as a multiple of g.