A ydiver falling from a great height is pulled toward the ground by the force of

Question

A ydiver falling from a great height is pulled toward the ground by the force of gravity (F_9 = mg). She also experiences a frictional force, air resistance, in the opposite direction which is proportional to the square of her speed (f_f= kv^2); k is a positive constant). If these are the only two forces she experiences, use Newton's Second Law to write a differential equation, in terms of her velocity, that expresses the net force acting upon her. Find any equilibrium solutions to the differential equation and classify them as stable, unstable or semi-stable. (you only need consider positive values of v.) c) Draw a phase diagram and sketch typical solutions of the equation. d) What is her terminal velocity (i.e. her speed as t rightarrowinfinity)?

Explanation / Answer

a) According to Newton's second law,

   Fnet = ma

   mg - kv^2 = ma

g - (k/m) v^2 = a

dv/dt = - (k/m) v^2 + g

c) dv/dt = g- (k/m) v^2

dv/[ 1- (k/mg) v^2 ] = gdt

At equilibrium, Fnet = 0

mg =kv^2

Vterm^2 = mg/k

so the integration gives

v(t) = Vterm tanh[gt/Vterm]

(arctanh is the inverse function of tanh defined as tanh = (e^z e^z )/(e^z + e^z ))

D) At equilibrium, Fnet = 0

mg =kv^2

Vterm^2 = mg/k

Vterm = sqrt ( mg/k)

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