Pleas solve this ASAP Let us try to derive one of the scaling laws from simple a

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Pleas solve this ASAP

Let us try to derive one of the scaling laws from simple arguments. We will try to derive the law for material ejection. Suppose that the dominant effect of the impact of a meteor is to throw material out of the crater. Assume that the material has a uniform density, rho, and the impact excavates a crater of radius, r, which is a perfect hemisphere, so the crater depth is the same as its radius. The material that is thrown out is lifted, on average, a height approximately equal to the crater depth. What is the volume of material thrown out of the crater during the impact? What is the mass of material thrown out of the crater during the impact? Once it has all landed (now being a typical distance, r, above its original height) what is the change of gravitational potential energy of the material thrown out of the crater? The scaling law we are looking for is an example of a "power law" relationship. A "power law" relationship is of the form y = Cxp, where p is some constant. So for example, the relationship between kinetic energy and speed (K = mv2/2) has p = 2. What is p in the power law relationship between energy and crater radius if material ejection dominates?

Explanation / Answer

1) 2/3*pi*r^3

2) rho*V = rho*2/3*pi*r^3

3) mgh = rho*2/3*pi*r^3*g*r = rho*2/3*pi*r^4

4) energy = C*radius^4

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