Solving for v: Seeking Guidance for Differential Equation

[samjohnny]

Homework Statement

Kindly see the attachment.

Homework Equations

The Attempt at a Solution

As with all such questions, its in setting everything up that I'm having some trouble.

I know that F = mdv/dt + vdm/dt. And also that F = R - m(t)g, but R = M0g. From here though I don't know how to proceed to obtain the differential equation to solve for v. Any hints?

Thank you very much.

 

[DEvens]

I know that F = mdv/dt + vdm/dt. And also that F = R - m(t)g, but R = M0g.

Careful there. Is the m in the first equation the same as the m in the second equation?

Also, you have been given m(t), right?

 

[jbriggs444]

How sure are you that F = m dv/dt + v dm/dt? Using principles of conservation of momentum in systems that are explicitly not closed is a tricky thing to get right.

In the frame of reference in which the balloon is momentarily at rest, v = 0 and the v dm/dt term goes away. Acceleration does not change depending on the choice of reference frame. It follows that F = m dv/dt or, more familiarly, F = ma.

 

[samjohnny]

Thanks for all the replies. I apologise for not getting back until now, my internet has been on the fritz. Ok, so I'm trying to be more careful with my m's. M is for the mass of the balloon, and m for the mass of the sand. So I have F = Mdv/dt = R - M(t)g, where R = M0g, and m(t) is the mass of the sand in the balloon. I calculated m(t) and got m(t) = m0(1 - t/T). Is that right so far? I'm just about to plug that into my force equation and I'll get back to you all on how that goes. Thanks

Update: Ok so in my force equation I have M(t)g, but I have only worked out m(t). Would it be valid to make the assumption that since the mass of the balloon << mass of sand, that their combined mass is approximately the mass of the sand. I.e. M ~ m?

 

[samjohnny]

Anyone?

 

[jbriggs444]

M is for the mass of the balloon, and m for the mass of the sand. So I have F = Mdv/dt = R - M(t)g, where R = M0g, and m(t) is the mass of the sand in the balloon. I calculated m(t) and got m(t) = m0(1 - t/T). Is that right so far?

Yes. This all looks correct.

Update: Ok so in my force equation I have M(t)g, but I have only worked out m(t). Would it be valid to make the assumption that since the mass of the balloon << mass of sand, that their combined mass is approximately the mass of the sand. I.e. M ~ m?

No. The stated problem is careful to distinguish between m and M. You must be equally careful.

 

[samjohnny]

Thanks for all the help!