The Starship Enterprise (variable acceleration problem) part II

[frankR]

Okay what am I doing wrong? This is the way I've been doing math for the last two years. This is annoying me. Unless I've been doing everything wrong the last two years, I feel this is correct. I realize it's most likely wrong. Someone please explain to me what I am doing wrong and more important why.

F = -be^(-a*v) = m dv/dt, a and b are constants.

m [inte]_vo^v e^(a*v) dv = -b [inte]_to=0^t dt

m/a e^(a(v - v_o)) = -b*t

ln[e^(a(v - v_o))] = ln[-abt/m]

a(v - v_o) = ln[-abt/m]

v(t) = 1/a ln[-abt/m] + v_o

dx/dt = v(t) = 1/a ln[-abt/m] + v_o

[inte]_xo=o^x dx = [inte]_to=o^t1/a ln[-abt/m] + v_odt

x(t) = t/a[ln(-a*b*t/m) + a*v_o -1]

 

[quantumdude]

Originally posted by frankR
m [inte]_vo^v e^(a*v) dv = -b [inte]_to=0^t dt

m/a e^(a(v - v_o)) = -b*t

Your mistake is in the last line here. When you do the integral, you have to evaluate exp(av) at v and at v₀ and subtract, to get:

exp(av)-exp(av₀).

This does not equal:

exp(a(v-v₀)),

which is what you have. Incidentally, this is the same basic mistake that I pointed out in "Part I" of this problem, except there you did it with the inverse function (natural log), when you used the invalid rule:

ln(a+b)=ln(a)+ln(b).

 

[frankR]

m/a e^(a*v)|_vo^v = -b*t

m/a(e^(a*v) - e^(a*v_o) = -b*t

Okay now that makes sense.

Thanks.

Edit: Hopefully I won't make that mistake again.