Training Resistance Parachute
Differential equations — fall of a paratrooper problem?
According to a newspaper account, a paratrooper survived a training jump from 1182 ft when his parachute failed to open but provided some resistance by flapping in the wind. Allegedly he hit the ground at 101.75 mi/h after falling for 6 seconds. To test the accuracy of this account, you should first find the drag coefficient , assuming a terminal velocity of 101.75 mi/h and also that the resistance of the paratrooper falling through the air is proportional to his velocity.
1. find the drag coefficient
2. distance fallen in 6 seconds
The units are metres and seconds.
If his mass is m and the drag coeff. is k and if we measure distance s downwards from his initial point, his equation of motion is md²s/dt² = mg − k.ds/dt or …
d²s/dt² + c.ds/dt = g where c=k/m.
Solution is sum of the Complementary Function and the Particular Integral.
The CF is Aexp(-ct) + B and the PI is gt/c so solution is s = Aexp(−ct) + gt/c + B
At t=0, s=0, ds/dt=0 so 0 = A+B and 0 = −cA+g/c → A = −B = g/c²
∴ s(t) = (g/c²){ exp(−ct) − 1 } + gt/c
Assuming he reached his terminal velocity when he hit the ground c.ds/dt = g
101.75 mph = 101.75 x 1609.34 / 3600 = 45.49 m/s.
This gives c = 9.81/45.49 = 0.2157 s‾¹ . Without mass you cannot work out k.
Using this value s(t) = 210.85{ exp(−0.2157t) − 1 } + 45.49t
So s(6) = 210.85{ exp(−6×0.2157) − 1 } + 45.49×6 = 119.88 m
This is very different to 1182 ft. I believe my arithmetic to be correct. If you put c=0 you get fall with no resistance and s=½ gt². Even this only gives 176.58 m.
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