Variable Mass Problem

Variable Mass Problem ...

Only minutes old ...
I have not even written the solution yet.

I get 1/k * ln 2 but I haven't checked just yet.

one of the two answers is

I get 1/k * ln 2 but I haven't checked just yet.

Bloody hell, how are you so good at mechanics?!

one of the two answers is

I get $\frac{1}{k}\ln 2$ as well, but I don't see where another answer is going to come from, so maybe I've just fluked it. I derived the eqns of motion from scratch though, so perhaps I screwed up. Roughly, for the raindrop, assuming infinitesimal changes of mass and velocity $dm, dv$ over time $dt$:

$dI = p_f-p_i=F dt \Rightarrow \\ \\ (m+dm)(v+dv)-mv = mg dt \Rightarrow m\frac{dv}{dt}+v\frac{dm}{dt}=mg$

But $m(t) = Me^{kt} \Rightarrow m\frac{dv}{dt}+kvm=mg \Rightarrow \frac{dv}{dt}+kv=g$

Thereafter it's just integration and initial conditions so I'm not sure what I've missed.

clearly too tired

the same equation
2 solutions

one as
a = dv/dt
the other
a = v dv/dx

OK, perhaps I'm being really dense but I have no idea what you are saying here. We can only get one time from that equation, AFAICS.

Didn't see the last bit aha, I'll work it out now
Edit: I get $x = \frac{g}{k^2}(\ln 2 - 0.5)$

the equation of motion should be after simplification

dv/dt = g - kv

solving this as it is gives you your answer.

you can also solve the same ODE as

v dv/dx = g - kv

to get v as a function of x

Bloody hell, how are you so good at mechanics?!

I was thinking the same thing..

Yes - see my edited answer above. You hid the second question too well. This is a pretty standard variable mass question though. Got any harder ones?