To check one of the MythBusters TV episodes, i.e., the fall of a mannequin (80 Kg) from a plane at an altitude of 1200 m in 31 s, with Maple13 (Windows Vista) I solved the following differential equation with initial conditions:

**> de:= m*(D@@2)(x)(t) = m*g - k*(D(x)(t))^2:**

**> ini:= (x(0) = 0, D(x)(0) = 0):**

> X:= unapply(rhs(expand(dsolve({de,ini}))),t);

X := proc (t) options operator, arrow; -m^(1/2)*g^(1/2)*t/k^(1/2)-m*ln(2)/k+m*ln(exp(2*g^(1/2)*k^(1/2)*t/m^(1/2))+1)/k end proc

1) Posing: *V0* = **sqrt**(*m* * *g* / *k*), *T* = **sqrt**(*m*/(*g***k*)) one has *V0***T* = *m*/*k*. I want to have:

> Xxb:= t -> V0*(- t + T * ln((exp(2*t/T) + 1)/2) ); # m.

How to obtain this equivalent equation *Xxb*(*t*), without retyping the equation of *X*(*t*) ? With **subs** , **convert** or **simplify** applied to *X*(*t*), Maple 13 gives error messages.

2) Taking the derivative of *Xxb*(*t*), one find :

V := proc (t) options operator, arrow; V0*(exp(2*t/T)-1)/(exp(2*t/T)+1) end proc.

How to transform it to *V* = *V0* tanh(*t*/*T*) ?

Again, **subs** , **convert** or **simplify** seem not working, even with **assume**(t, real), **assume**(T, real) ! I know simplify is a difficult task, but Maple should recognize a tanh !

Thank you in advance for any suggestion.

( Note that with *m* = 80 Kg, *g* = 9.81 m/s^2 and *k* = 0.267482 Kg/m, which correspond to a speed limit of *V0* =195 km/h, on find *t*(1200m) = 26 s, instead of 31 s ).