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Samantha24
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Homework Statement
A free-falling sky-diver of mass M jumps from an aeroplane and beforen he opens his parachute experiences air resistance which is proportional to the square of the magnitude of his velocity.
a) Show that the equation of motion for the sky-diver can be written as
dv/dt = -g + (k/M) v^2
where v is the magnitude of the sky-diver's velocity, k is the proportionality constant for the air resistance and g is the gravitational constant.
b) What initial data concerning, v(0), is consistent with the following solution
the of differential equation obtained in part (a) of this question,
v(t) = square root ((Mg/k) ((e^-At - 1/ e^-At + 1)) ;
where
A = 2* (square root gk/M)
c) Using the expression for the velocity obtained in part (b) of this question, show that there exists a limiting or terminal velocity, vL, such that
v(t) tends to vL, t tends to infinity.
What is vL?
Homework Equations
dv/dt = -g + (k/M) v^2
v(t) = square root ((Mg/k) ((e^-At - 1/ e^-At + 1)) ;
where
A = 2* (square root gk/M)
The Attempt at a Solution
a) done
b)
dv/[(k/m)v^2 - g] = 1
sqrt(m/kg)arctan(v sqrt k/mg) = t + c
arctan(v sqrt k/mg) = sqrt(kg/m)*(t+C)
v sqrt(k/mg) = tan[sqrt(kg/m)*(t+C)]
v(t) = sqrt(mg/k)tan[sqrt(kg/m)*(t+C)]
c)
vL = lim{t→∞} √(mg/k)*(e^(-At) - 1)/(e^(-At) + 1) = √(mg/k)
I can't get the exponentials. Help appreciated.
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