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LANS

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

A spherical asteroid of mass [tex]m_{0}[/tex] and radius R, initially moving at speed [tex]v_{0}[/tex], 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 [tex]\frac{dv}{dt} = -kv^{3}[/tex] and evaluate k.

b) find [tex]v(t)[/tex]

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

Conservation of momentum:

[tex]m_{0}v_{0} = m(t) v(t)[/tex]

[tex]dm = (\pi R^2)*(v(t)dt)*D[/tex]

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

**Homework Statement**A spherical asteroid of mass [tex]m_{0}[/tex] and radius R, initially moving at speed [tex]v_{0}[/tex], 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 [tex]\frac{dv}{dt} = -kv^{3}[/tex] and evaluate k.

b) find [tex]v(t)[/tex]

**The attempt at a solution**Let [tex]A_{c} = 2*\pi*R[/tex] be the cross-sectional area of the asteroid.

Conservation of momentum:

[tex]m_{0}v_{0} = m(t) v(t)[/tex]

[tex]dm = (\pi R^2)*(v(t)dt)*D[/tex]

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

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