an object traveling horizontally through air experiences quadratic drag and we c

Question

an object traveling horizontally through air experiences quadratic drag and we can model it using the expression a=-fv^2 Where f is a positive constant. Suppose an object has an initial velocity v(0)=vi and initial position x(0)=0.
a) what are the units of the constant f? b) derive a simplified expression for v(t) using the initial condition given above. c) derive an expression for x(t) using the initial conditions given above. d) what is the lim x(t) as t approaches infinity? an object traveling horizontally through air experiences quadratic drag and we can model it using the expression a=-fv^2 Where f is a positive constant. Suppose an object has an initial velocity v(0)=vi and initial position x(0)=0.
a) what are the units of the constant f? b) derive a simplified expression for v(t) using the initial condition given above. c) derive an expression for x(t) using the initial conditions given above. d) what is the lim x(t) as t approaches infinity? an object traveling horizontally through air experiences quadratic drag and we can model it using the expression a=-fv^2 Where f is a positive constant. Suppose an object has an initial velocity v(0)=vi and initial position x(0)=0.
a) what are the units of the constant f? b) derive a simplified expression for v(t) using the initial condition given above. c) derive an expression for x(t) using the initial conditions given above. d) what is the lim x(t) as t approaches infinity?

Explanation / Answer

a)

units of f are m-1

b)

a= dv/dt = -fv2

so -1/fv2 dv = dt

so v(t) = 1/(ft + c1) where c1 is a constant

c ) v = dx/dt

so x(t) = integral (v(t)) = ln(ft+c1) / f + c2

using the initial conditions.... v(0) = vi = 1/c1 ==> c1 = 1/vi

so V(t) = 1/(ft + (1/vi)) = vi/(1+ft*Vi )

d) x(0) = 0 ==> c2 = -ln(c1) /f = ln(Vi) /f

so X(t) = ln(ft+ (1/vi) ) / f + ln(Vi)/f = ln(ft*Vi + 1) / f

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