- #1
Gwilim
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Homework Statement
A vehicle of mass m experiences a constant frictional resistance ma and air resistance proportional to the square of its speed. It can exert a constant propelling force mb and attain a maximum speed V Show that, starting from rest, it can attain the speed V/2 in the time
Vln3/2(b-a)
And that the friction and air resistance alone can then bring it to rest in a further time
(V/(a(b-a))^1/2)tan-1(b-a/4a)^1/2
2. The attempt at a solution
Using mc as the constant of proportionality for air resistance yields:
x''=b-a-c(x')^2
from there I'm just guessing as to the method but I've tried this:
dx'/dt=b-a-c(x')^2
int(dx')=int((b-a-c(x')^2)dt)
=(b-a)int(dt)-c(int((x')^2dt)
=(b-a)int(dt)-c(int(x'dx))
but I'm not sure how to integrate x' w.r.t. x?
Of course I could be going about it completely the wrong way. The motion is in one dimension so I could maybe import one of the equations of motion, like v^2=u^2+2as? I'm sure I'm missing a trick (or two) somewhere.