Compute the velocity of a free-falling parachutist using Euler's method

[thornear]

Homework Statement

For the second order drag model (Eq. 1.8), compute the velocity of a free-falling parachutist using Euler's method for the case where,

m = 80 kg
C_d = .25 kg/m

Perform the calculation from t = 0 to 20 with a step size of 1 s. Use an initial condition that the parachutist has an upward velocity of 20 m/s at t = 0. At t=10 s, assume that the chute is instantaneously deployed so that the drag coefficient jumps to 1.5 kg/m.

Homework Equations

Eq. 1.8,

dv/dt = g-((C_d)/m)*v²

The Attempt at a Solution

Used equation v(t_i+1) = v(t_i) + [g - (C_d/m)*v(t_i)²](t_i+1 - t_i)v

(Used in example in book, unfortunately no example w/ an initial condition with an "upward velocity" though)

Plugged in the values to achieve,

t = 0... V = 20 + [9.81 - (.25/80)(0)²] *1 = 29.81m/s
t= 1...V = 29.81 + [9.81 - (.25/80)²]*1 = 36.51m/s
t =2....V=36.51 + [9.81-(.25/80)²]*1 = 42.15 m/s
...so on until t = 10 where C_d changes from .25 to 1.5

Am I doing this right? I don't know how the "upward velocity = 20" works into this. I assumed that it is the initial v(t_i) as you can see from the first solution I have where t = 0, which may or may not be horribly wrong.

Thanks, much appreciated.

 

[D H]

Since you are representing an upward velocity as positive, you must have g as negative here. Try making g negative.

 

[thornear]

Making g negative makes sense to me but it begins to produce negative velocities. Also I made a mistake in the above post,

t = 0... V = 20 + [9.81 - (.25/80)(0)2] *1 = 29.81m/s
t= 1...V = 29.81 + [9.81 - (.25/80)2]*1 = 36.51m/s
t =2....V=36.51 + [9.81-(.25/80)2]*1 = 42.15 m/s
...so on until t = 10 where Cd changes from .25 to 1.5

Should read

t = 0... V = 20 + [9.81 - (.25/80)(0)²] *1 = 29.81m/s
t= 1...V = 29.81 + [9.81 - (.25/80)(29.81)²]*1 = 36.51m/s
t =2....V=36.51 + [9.81-(.25/80)(36.51)²]*1 = 42.15 m/s
...so on until t = 10 where Cd changes from .25 to 1.5

But in looking at that I realize perhaps that v(t_i)² should be 20² initially rather than 0².

Thoughts? Help?

Thanks

 

[D H]

Let's step back and think about the math here, and the physics as well.

You initially have the parachutist going upwards. Gravitational acceleration is obviously directed downwards. You will need some sign convention on velocity and acceleration to handle this scenario. If you choose the sign convention such that downward velocity is positive, your value for g will be positive and your initial velocity will be negative. If you choose the sign convention such that upward velocity is positive, your value for g will be negative and your initial velocity will be positive.

That is not enough to cover this scenario. The other problem is drag. Drag is always directed against the velocity vector. If your velocity is positive (regardless of sign convention), the drag acceleration must be negative. If your velocity is negative, the drag acceleration must be positive. You have drag acceleration as $- (c_d/m)v^2$, so it is always negative. This is incorrect for negative velocities. One way to correct this is to compute the drag acceleration as $- (c_d/m)(v^3/|v|)$.

 

[rwisz]

Your approach using Euler's method seems correct. The initial upward velocity of 20 m/s can be accounted for by using it as the initial condition for v(ti) at t = 0. This means that at t = 0, the parachutist has an initial velocity of 20 m/s upwards, and then starts to free-fall under the influence of gravity and drag.

One thing to note is that in the equation for Euler's method, the term (ti+1 - ti) should be replaced with the step size, which in this case is 1 s. So the equation should be v(ti+1) = v(ti) + [g - (Cd/m)*v(ti)2]*1.

Also, at t = 10 s, the drag coefficient jumps to 1.5 kg/m, so this should be taken into account in the calculation for v(ti+1) at t = 10 s and onwards.

Overall, your approach is correct and you have accounted for the initial upward velocity correctly. Keep in mind any changes in the parameters (such as the drag coefficient) during the calculation.