Momentum conservation of asteroid in a dust cloud

[LANS]

Note: this is one of the suggested practice problems for my second-year classical mechanics course.

Homework Statement
A spherical asteroid of mass $$m_{0}$$ and radius R, initially moving at speed $$v_{0}$$, encounters a stationary cloud of dust. As the asteroid moves through the cloud, it collects all the dust that it hits, and slows down as a result. Ignore the increase in radius of the asteroid, and its gravitational effect on distant dust grains. Asume a uniform average density D (mass per unit volume) in the dust cloud.

a)show that $$\frac{dv}{dt} = -kv^{3}$$ and evaluate k.
b) find $$v(t)$$

The attempt at a solution

Let $$A_{c} = 2*\pi*R$$ be the cross-sectional area of the asteroid.

Conservation of momentum:
$$m_{0}v_{0} = m(t) v(t)$$
$$dm = (\pi R^2)*(v(t)dt)*D$$
dm is from mass of dust which the asteroid hits in time dt. Cross-sectional area * distance traveled in time dt * dust density.

I'm not sure where to go from here. Any help is greatly appreciated.

Thanks

 

[diazona]

$$m(t) = m_{0} - dm*t$$

Where did this equation come from? It doesn't really make sense (and besides, the units are inconsistent, so at least that needs to be fixed).

$$dm = (2 \pi R)*(v(t)dt)*D$$
dm is from mass of dust which the asteroid hits in time dt. Cross-sectional area * distance traveled in time dt * dust density.

What's the cross-sectional area of a sphere? Remember that it has to have the proper units for area

 

[LANS]

Fixed first post.

 

[LANS]

Where did this equation come from? It doesn't really make sense (and besides, the units are inconsistent, so at least that needs to be fixed).

$$m(t) = m_{0} - dm*t$$

Thanks, didn't quite think about that one. Forgot to consider that the amount of mass the asteroid picks up in a set amount of time is dependant on its velocity, and is thus not a constant term.

 

[diazona]

You should respond in your new post rather than going back and changing what you previously posted - not only because it makes it possible for others to follow the discussion, but also because editing your original post after it's been replied to is a violation of the PF guidelines.

Anyway, what is the value of this expression?
$$\frac{\mathrm{d}(mv)}{\mathrm{d}t}$$
The resulting equation will be useful to you. Together with the two that are currently in your original post, it will allow you to solve the problem.

 

[Mindscrape]

Alternatively (really the same as from diazona in the end, but slightly different initially), note that m = m0v0/v, so what's dm in terms of dv?

From part a) onwards you just have to solve the simple ODE.