I already see "no solution" for the i2 expression alone, see below.
And if I enter this in Mathematica.
e = (-g*m^2 - Vy*k*m)*Exp[-k*t/m] - g*k*m*t + g*m^2 + Vy*k*m
Solve[e == 0, t]
I get:
===================================================================
During evaluation of In[2]:= Solve::ifun: Inverse functions are being
used by Solve, so some solutions may not be found; use Reduce for
complete solution information.
Out[2]= {{t -> (
g m + k Vy +
g m ProductLog[-((E^(-1 - (k Vy)/(g m)) (g m + k Vy))/(g m))])/(
g k)}}
https://reference.wolfram.com/language/ref/ProductLog.html
===================================================================
Maple gives:
solve((-g*m^2 - Vy*k*m)*exp(-k*t/m) - g*k*m*t + g*m^2 + Vy*k*m, t);
0
===================================================================
I don't know wether this makes you happy or sad.
Ralf
===================================================================
(1) -> v:=operator 'v
(1) v
Type:
BasicOperator
(2) -> equ := m*a=-m*g-k*v(t)
(2) a m = - k v(t) - g m
Type:
Equation(Expression(Integer))
(3) -> equ1 := subst(equ, a=D(v(t),t))
,
(3) m v (t) = - k v(t) - g m
Type:
Equation(Expression(Integer))
(4) -> sol := solve(equ1, v, t=0, [V0])
k t
- ---
m
(g m + V0 k)%e - g m
(4) -------------------------
k
Type:
Union(Expression(Integer),...)
(5) -> i := integrate(sol, t=0..z)
k z
- ---
2 m 2
(- g m - V0 k m)%e - g k m z + g m + V0 k m
(5) --------------------------------------------------
2
k
Type: Union(f1:
OrderedCompletion(Expression(Integer)),...)
(6) -> i1 := subst(i, [V0=Vx, g=0]) = s
k z
- ---
m
- Vx m %e + Vx m
(6) --------------------- = s
k
Type:
Equation(Expression(Integer))
(7) -> i2 := subst(i, V0=Vy) = 0
k z
- ---
2 m 2
(- g m - Vy k m)%e - g k m z + g m + Vy k m
(7) -------------------------------------------------- = 0
2
k
Type:
Equation(Expression(Integer))
(8) -> h := solve([i1,i2], [s, z])
(8) []
Type:
List(List(Equation(Expression(Integer))))
(9) -> h1 := solve([i1], [s, z])
- k s + Vx m
m log(------------)
Vx m
(9) [[z = - -------------------]]
k
Type:
List(List(Equation(Expression(Integer))))
(10) -> h2 := solve([i2], [z])
(10) []
Type:
List(List(Equation(Expression(Integer))))
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