Spherical asteroid moving through a dust cloud

• Kyliticus
In summary, the problem deals with a spherical asteroid moving through a stationary dust cloud and collecting dust grains as it moves. The velocity of the asteroid is shown to obey the equation a = -k*v^3 by considering momentum conservation over an infinitesimal time interval, and the constant k is evaluated. The velocity of the asteroid as a function of time is also determined. The solution involves manipulating equations and considering the hyperbolic motion of the asteroid.
Kyliticus

Homework Statement

A spherical asteroid of mass m0 and radius, R, initially moving at speed v0, 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. Assume a uniform average density D (mass per volume) in the dust cloud. By considering momentum conservation over an infinitesimal time interval dt, show that the velocity v of the asteroid obeys a = -k*v^3, and evaluate constant k. Also, find the velocity of the asteroid as a function of time.

Homework Equations

a = -k*v^3 (must evaluate for k)

The Attempt at a Solution

I am at a complete loss for this question. I recall in class we did an example with a rocket and its fuel, where the rocket loss mass as the fuel it had expelled gained mass. I know this question deals with some of the same concepts, but I really need a push to get in the right direction. Thanks!

I've got an answer to this.
This problem caught my attention as every problem linked to motion and speed.
I have troubled some times, but in the end, manipulatin the equations, I came to the formula involving the $$v^3$$. I have even done a simulation of this dust collector to convince myself that the acceleration goes with the cube of the speed.
The explanation is quite simple, of course, after you've found it :)
I'll give it to you right now, I don't like to ping-pong over.
Anyway, ...

you've got this asteroid moving, whose momentum is $$mv = k = m_0v_0$$.
Since no force acts on the object, k is a constant.
So, differentiating the expression yelds:

$$dm\ v\ +\ m\ dv = 0$$

$${dm\ \over m}\ = -{dv \over v}$$

And, will be useful later

$${dm\ \over dt}{1 \over m}\ = -{dv \over dt}{1 \over v}$$

Basically what we have is an hyperbola, now we have to find how the point moves with time over this hyperbola.

In a time $$dt$$ we can think $$m$$ and $$v$$ as constants and the object collects a mass of dust $${dm } = v DS dt$$ where $$DS$$ is the density of dust per volume multiplied by the surface normal to velocity.
Then

$${dm \over dt} = v DS$$

To find an expression of $$dv/dt$$ which contains only speed as a variable (the hard parte for me), we remember that

$${dm\ \over dt}{1 \over m}\ = -{dv \over dt}{1 \over v}$$

or

$${dv \over dt} = -{dm\ \over dt}{v \over m}\$$

In the expressions above we find formulas to replace $$m$$and $$dm/dt$$

which give us finally

$$a = {dv \over dt} = -{v DS}\ v\ {v \over m_0v_0}\ = -v^3 {DS \over m_0v_0}$$

The expression of speed in function of time should be.

$$v = \sqrt { {m_0v_0^2} \over {m_0+2v_0DSt}}$$

Any comment or objection is welcome.

Last edited:
Absolutely amazing explanation Quinzio, a lot more work shown than I would have expected too! I'm definitely going to take an hour to review this thoroughly, thanks a ton.

What is a spherical asteroid?

A spherical asteroid is a celestial object that has a nearly round shape, similar to a ball. It is composed of rock, metal, and other materials and can range in size from a few meters to hundreds of kilometers in diameter.

How does an asteroid move through a dust cloud?

An asteroid moves through a dust cloud by following the laws of motion and gravity. As the asteroid approaches the dust cloud, the particles in the cloud will exert a force on the asteroid, causing it to change its direction and speed.

What is the composition of a dust cloud?

A dust cloud is typically composed of small particles of dust, ice, and gas. The exact composition can vary depending on the location and source of the cloud.

Can a spherical asteroid get trapped in a dust cloud?

Yes, it is possible for a spherical asteroid to get trapped in a dust cloud. This can happen if the asteroid's trajectory takes it through the densest part of the cloud or if the asteroid's velocity is not high enough to escape the gravitational pull of the cloud.

What effects can a dust cloud have on a spherical asteroid?

A dust cloud can have several effects on a spherical asteroid, including altering its trajectory, causing it to heat up and potentially break apart, and changing its surface appearance due to dust accumulation. The effects will depend on the size and composition of the asteroid as well as the density and composition of the dust cloud.

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