5.2.1 The solution for a downwardly moving sphere
5.2.1 The solution for a downwardly moving sphere
Clear[v,m,g,k,velocity2down]
vsolution2down =
DSolve[{m v'[t] == -m g - k v[t]^2,v[0] == v0}, v[t], t]
Out[30]=
{{v[t] -> (Sqrt[g] Sqrt[m]
Tan[Sqrt[g] Sqrt[k]
Sqrt[k] v0
ArcTan[---------------]
t Sqrt[g] Sqrt[m]
(-(-------) + -----------------------)]) / Sqrt[k]}}
Sqrt[m] Sqrt[g] Sqrt[k]
Solve::ifun:
Warning: Inverse functions are being used by Solve, so
some solutions may not be found.
Solve::ifun:
Warning: Inverse functions are being used by Solve, so some
solutions may not be found.
In[31]:=
velocity2down = v[t]/.vsolution2down[[1]]
Out[31]=
(Sqrt[g] Sqrt[m] Tan[Sqrt[g] Sqrt[k]
Sqrt[k] v0
ArcTan[---------------]
t Sqrt[g] Sqrt[m]
(-(-------) + -----------------------)]) / Sqrt[k]
Sqrt[m] Sqrt[g] Sqrt[k]
Integrating to get y[t],
In[32]:=
ysolution2down =
DSolve[{y'[t] == velocity2down, y[0] == y0}, y[t], t]
Out[32]=
Sqrt[k] v0
m Log[Cos[Sqrt[k] ArcTan[---------------]]]
Sqrt[g] Sqrt[m]
{{y[t] -> y0 - ------------------------------------------- +
k
(m Log[Cos[Sqrt[g] Sqrt[k]
Sqrt[k] v0
ArcTan[---------------]
t Sqrt[g] Sqrt[m]
(-(-------) + -----------------------)]]) / k}}
Sqrt[m] Sqrt[g]
In[33]:=
height2down = ysolution2down[[1,1,2]]
Out[33]=
Sqrt[k] v0
m Log[Cos[Sqrt[k] ArcTan[---------------]]]
Sqrt[g] Sqrt[m]
y0 - ------------------------------------------- +
k
(m Log[Cos[Sqrt[g] Sqrt[k]
Sqrt[k] v0
ArcTan[---------------]
t Sqrt[g] Sqrt[m]
(-(-------) + -----------------------)]]) / k
Sqrt[m] Sqrt[g]