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Hedrix

Air Resistance

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Ok guys, I'd like you to do me a favour ^^ please. I've been trying to solve one physics exercise(for university studies...), however I couldn't get the correct results. This is it: A man in a parachute falls from rest and reaches a limit velocity of 60 m/s. How long will it take him to reach 30 m/s? Assume that air resistance is directly proportional to the velocity and take g = 9.8 m/s2. At first, it seemd simple, but when I started working on it, and didn't get correct answer...I'll just tell what I've tryed so far: I don't know physics terminology good enough in english, so sorry for that ^^ 1.)I know that this object is being affected by 2 forces, Gravity force,I'll name it W, and Air resistance force, I'll name it D (as somtimes Air resistance force is called Drag Force). 2.)v=30m/s; v1=60m/s; When obejct is at velocity of 30 m/s, his acceleration a=(W-D)/m; thus t=v/(W-D)/m; because W=mg, we can write t=v/(mg-D)/m; When obejct is at velocity of 60 m/s (so called terminal velocity or limit velocity), a1=0 and W1=D1; so, now real fun begins >< if W1=W (as it's the same object) and D=D1/2 (or not?). I assumed that because v=v1/2; as for further...I really am not sure and that's why I need some help. Please share with me your wisdom and help a newcomer to university^^

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Quote:
Original post by Hedrix
2.)v=30m/s; v1=60m/s;
When obejct is at velocity of 30 m/s, his acceleration a=(W-D)/m; thus t=v/(W-D)/m; because W=mg, we can write t=v/(mg-D)/m;

ERROR1:
Quote:
Original post by Hedrix
When obejct is at velocity of 30 m/s, his acceleration a=(W-D)/m;

No! it's acceleration is always: a=(W-D)/m;

ERROR2:
Quote:
Original post by Hedrix
thus t=v/(W-D)/m;

No! a=dv/dt that is a=v' (acceleration is the derivative -wrt time- of velocity)

ERROR 3:
Quote:
Original post by Hedrix
thus t=v/(W-D)/m; because W=mg, we can write t=v/(mg-D)/m;

If you assume v'=v/t (wich is absolutey wrong) you should get t=v*m/(m*g-D)

Anyway!
Abstraction is good! Forget the numerical values. They are just for the numerical application.
Try first to find the mathematical model of your problem! that is: What are the variables and how they depend on each other.

Some hints (may or may not be correct... hehe!)
Begining with:
a=(W-D)/m; that is v'(t)=(W-D(v(t)))/m; (because acceleration and velocity are function of time and Drag is function of velocity.)
and:
D(v)=Constant*v; (D is proportional to v)

You should end up with a differential equation . a linear one (something like: c0*v+c1*v'+c2=0;), so v(t) have an exponential term (something like: v(t)=v_max*(1-exp(-(g/v_max)*t)); gess why! why doesn't it depend on the mass?).

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