A particle moves in a medium under the influence of a retarding force equal to m

Question

A particle moves in a medium under the influence of a retarding force equal to mk(v^3+a^2v) where k and a are constants. Show that for any value of the initial speed the particle will never move a distance greater than Pi/2ka and that the particle comes to rest only for t-->infinity.
Apply Newton's second law. Replace acceleration (a) with (v dv/dx), separate variables and integrate to get x=x(v). What is the largest x can be? Go back to the newtons law equation, replace a with (dv/dt). Separate and integrate. At what time(s) is v=0? A particle moves in a medium under the influence of a retarding force equal to mk(v^3+a^2v) where k and a are constants. Show that for any value of the initial speed the particle will never move a distance greater than Pi/2ka and that the particle comes to rest only for t-->infinity.
Apply Newton's second law. Replace acceleration (a) with (v dv/dx), separate variables and integrate to get x=x(v). What is the largest x can be? Go back to the newtons law equation, replace a with (dv/dt). Separate and integrate. At what time(s) is v=0?
Apply Newton's second law. Replace acceleration (a) with (v dv/dx), separate variables and integrate to get x=x(v). What is the largest x can be? Go back to the newtons law equation, replace a with (dv/dt). Separate and integrate. At what time(s) is v=0?

Explanation / Answer

F = mk(v^3 + a^2*v)

So,

a = F/m = k(v^3 + a^2*v)

Now,

a = v*dv/dx

So, v*dv/dx = k*(v^3+a^2*v)

So, dv/dx = k*(v^2 + a^2)

So, dv/(v^2 +a^2) = k*dx

So, by integrating both sides, we get

atan(v/a)/a = k*x + P <------ P = some constant

Let the initial velocity be u = 0

So, at x = 0 , atan(u/a)/a = P

So, u = tan(Pa)*a = 0

So, P = 0

So, so, for maximum distance covered, Xmax ,

maximum value of atan(v/a) = Pi/2

So, k*Xmax = Pi/2a

So, Xmax = Pi/2ka <--------answer(Proved)

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