I need help solving this.. please?

Question

If a can begins empty, sitting at rest in a sea of pressurized air and the valve is opened, then what would happen to the can?
Wil the can move to the left, right, or not move once the valve is opened and air is let inside

Make the following assumptions: Pressure outside can = P ; mass of can = m; Area of can ends(each) = A; area of valve hole = s.

Newton's second law :

m*a = P*s*e^(-bt) - C*v(t)

a = d^2x / dt

v = dx / dt

m*d^2x / dt = P*s*e^(-bt) - C*dx / dt

m*d^2x / dt + C*dx / dt = P*s*e^(-bt) >>> Differential equation

the solution for this differential eq is:
y’ = Ps/(mb-c) { e^ (- c/m * t) - e^ (- bt)}

How can i show that (mb-c) and { e^ (- c/m * t) - e^ (- bt)} are both positive?
granted, that we say positive=the right direction

the actual question is on:
http://answers.yahoo.com/question/index;_ylt=AuXbSuiFQmy84jsLSUidAuPty6IX?qid=20070601130745AAK5cl8

and the way it was solved is on:
http://answers.yahoo.com/question/index;_ylt=Aq5ZO0qSlr2b85jBBkEdNjPty6IX?qid=20070603120520AATkArz

Answer 1

With your equation, you can say that P is the pressure from outside on the left side area equal to the valve hole, which is directed to the right. Therefore, for the equations purpose it is a vector.

You do not, as you expressed, want to prove that mb-c and e^(-ct/m)-e^(-bt) are both positive. You simply want to prove that they are OPPOSITE, so that the overall equation is positive. This is relatively simple:

If e^... is to be positive, the first term must be greater than the second (since e^anything is always positive). Therefore, you need to prove that c/m is less than b (and therefore greater when multiplied by -t). Going from there:

c/m < b
c < mb (we know that m, c, and b are all positive)
0 < mb-c

So mb-c is positive also, and since the numerator and denominator are positive, the overall equation is positive.

However, if e^... is to be negative, the first term must be less than the second, and you need to prove that c/m is greater than b (and therefore less when multiplied by -t). Going from THERE:

c/m > b
c > mb
0 > mb-c

Therefore, the numerator and denominator are both negative, so the overall equation is still POSITIVE.

Since you know that the force acting on the can (from pressure) is directed to the right, you can prove that the velocity of the can is directed in the same direction.

Hope that helped. Have a good day.

Answer 2

I remember to have answered this a few minutes ago. Pl. think of the can as a small rocket motor with the air jet going inside the can from right to left and thus pushing the can from left to right.

Answer 3

It will no longer remain as a can.

Answer 4

stuff