※ 引述《newtype2007 (無業遊民)》之銘言：
: m*dv/dt=mg-av^2 v(0)=0
: m,g,a均為常數
: 實在是解不出來，拜託大家了!
m*dv/dt = mg - av^2
dv/dt = g - (a/m)v^2
dv/[g - (a/m)v^2] = dt
dv/[1 - (a/mg)v^2] = gdt
____
dv/[1 - ( √a/mg v )^2] = gdt
____ ____ ____
d ( √a/mg v) / [1 - ( √a/mg v )^2] = ( √ag/m ) dt
____
Let √a/mg v = x
-1
∫1/(1-x^2)dx = (1/2) ㏑| (1+x)/(1-x) | + C = tanh x + C
-1 ____ -1 ____ ____ ____
tanh [ √a/mg v(t) ] - tanh [ √a/mg v(0) ] = √ag/m ( t - 0 ) = √ag/m t
-1 ____ ____ -1 ____
tanh [ √a/mg v(t) ] = √ag/m t + tanh [ √a/mg v(0) ]
____ -1
= √ag/m t + tanh [ √a/mg 0 ]
____ -1
= √ag/m t + tanh [ 0 ]
____
= √ag/m t + 0
____
= √ag/m t
____ ____
√a/mg v(t) = tanh [ √ag/m t ]
____ ____
v(t) = √mg/a tanh [ √ag/m t ]
____
when t → ∞ => v(t) → √mg/a
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