Hard Variable Mass Problem

Here's my solution

I got as far as deriving the eqns of motion before I saw your full solution (you work too fast :-)) so I won't bother to finish off - my approach was a little different though, since you can work in terms of $v(x)$ from the start

$\displaystyle v(x)= \frac{dx}{dt} \Rightarrow dI = F dt = F \frac{dx}{v}$

Then we have by the usual argument:

$\displaystyle m dv + v dm = F \frac{dx}{v} \Rightarrow m v \frac{dv}{dx} + v^2 \frac{dm}{dx} = F$

and since $m(x)=M(\lambda x+1) \Rightarrow \frac{dm}{dx} =M\lambda$ and the external force is the weight, acting in the negative sense to $x$, then

$\displaystyle M(\lambda x+1) v \frac{dv}{dx} + v^2 M\lambda = -M(\lambda x+1)g \Rightarrow v \frac{dv}{dx} + \frac{\lambda v^2}{\lambda x+1} = -g$

that is my approach but the ODE will be tough for most students to solve ...

From which we can see that "A Slice of Pi" is not like most students.

I am stuck with this





Your help is valued and appreciated ...
@comoving_frame
@constellarknight

Honoured that you'd ask but I don't do mechanics. Anyway shouldn't you be fixing all the mistakes in your practice papers and topic booklets?

Here's my solution

That was what I did!!

Though @TeeEm, did you really not know how to do this? Or is it just to get people to try it?

As they are free I do not think so ...
Besides anybody with half a molecule of brain should be able to deal with any typos/errors by cross referencing between the typed question and the worked solution.

I hope I know how to do it ... but I feel better to "be one of us" and struggle to do it ...

Linux is free but that doesn't mean the maintainers fix bugs only out of grace: they do so because they know that if they behave in an arrogant and sanctimonious manner, people will eventually have had enough and will fork the project.

Similarly I have already begun creating competition to your resources with my Advanced Mathematics Extension Paper:
http://www.thestudentroom.co.uk/showthread.php?t=3905787. You think that you're so clever, and beautiful, and whatever, because you have 18,000 posts, which actually doesn't mean ****.

have you taken your medication today?

What medication? Again you are libelling me.

I would love to talk to you kid but I am a bit busy .... Maybe tomorrow ...

Why would you want to talk to me? You clearly have nothing non-frivolous and non-libellous to say.

If you have nothing relevant to the question to add to this thread, then why even bother to post?