The energy of debris ejected from an asteroid impact

[QuantumX]

I have been struggling with this problem. I'll greatly appreciate someone pointing me in the right direction. I think it's safe to assume the velocity of the impact is the Earth's escape velocity 11.1*10^3 m/s

Here's the problem:

Impacting objects will excavate a certain amount of debris material from a crater md and loft it into
Earth's atmosphere to a height h. For an impacting object of mass mi, calculate how much debris mass md is lofted to an altitude h (where it will have zero velocity due to deceleration in Earth’s gravity). Assume that h << Rearth. Your answer will depend on mi, h and Rearth. For simplicity, assume the impactor comes in from straight up (along a radial line) and the debris is lofted straight out, also on a radial line. Consider h = 10 km, which is the altitude of the Earth’s jet streams. If debris reaches this altitude it will circulate globally. How much mass will an asteroid with density 2 g/cm^3 and diameter 1km be able to lift to a 10 km altitude?

 

[mfb]

Well, a realistic model would be way beyond the scope of this homework, so I guess you are supposed to calculate the maximal amount of material, based on the impact energy.

It would be useful to know some context of the problem (in which course/class are you? What is the current topic?), but it looks like introductory physics.

 

[QuantumX]

It's not supposed to be that realistic, hence the radial trajectories. The course is an intro astrophysics class, and the context of the question is asteroid impacts.

If you consider it an intro problem, then please move it to the appropriate section (sorry, I'm new).

Thanks.

 

[BvU]

Would you be happy with a calculation where the kinetic energy of the asteroid is converted to potential energy of the debris? As if m_i fell on a seesaw with m_d at the other end ?

 

[QuantumX]

Sure, that sounds right. Although, wouldn't that be be kinetic energy being transferred from one body to the other?

I figured the mass of the asteroid would be the same as the mass of the ejected debris, and the energy would be the same as well. Where I get stuck is how to make that energy dependent on the height h and Earth's radius. I'm thinking Earth's gravity would have factor in somewhere, since h is dependent on it. And I assume geometry will be involved, since both trajectories are just straight radial lines, but I just can't figure out how to do it. Your approach sounds like the right track

 

[BvU]

If m_d = m_i there is nothing to calculate any more... The question - rephrased - is to calculate how much m_d you can lift 10 km with the kinetic energy in m_i at 1.11 x 10⁴ m/s. It is a rough upper limit, but probably not even all that bad.

 

[Chestermiller]

The average speed of asteroids is about 25 km/sec. That's more than 4x the kinetic energy of 11 km/sec.

Chet

 

[QuantumX]

@BvU Now that you put it this way it suddenly makes sense. Yes, then that is what I need! Thanks.

 

[QuantumX]

Ok, so based on BvU's recommendation that I transfer kinetic energy to potential energy.

Here is what I did:

Ed = md*h*g = md*h*G*Mearth/Rearth^2 (g = GMearth/r^2)

Ei = 0.5*mi*v^2 (v = escape velocity of Earth)

Ed = Ei

so I equate the two and solve for md and the answer depends on mi, h and Rearth, like the problem asked.

Did I do it right?

 

[mfb]

Looks good.