This is what Einstein would have called a gedanken experiment – one that can be

Question

This is what Einstein would have called a gedanken experiment – one that can be done in your mind, but not in practice. You are probably aware of the large impact crater found in the Chicxulub area of the Yucatan peninsula (Mexico). The crater was generated by the impact of a fairly large body some 65 million years ago, at the boundary between the Cretaceous and the Tertiary periods of geological history. This break in Earth’s history is marked by a mass extinction, when as many as half the species on the planet became extinct and, among these, the dinosaurs. For tens of millions of years these creatures had flourished. Then, suddenly, they disappeared. Particle physicist Luis Alvarez hypothesized that this extinction was brought about by the enormous amount of dust and ashes that permeated the atmosphere for a length of time as the result of the collision of a large body with the Earth. A search was initiated that brought to the discovery of this crater through satellite images of the area in 1980. The crater has a diameter of about 180 km, which leads us to believe the size of the impactor to have been about 10 km in diameters (the crater is usually more than ten times the size of the impactor). Since the impactor could have been a comet, a stony asteroid or a metallic asteroid, we estimate a mass range between 1.0 x 1015 kg and 4.6 x 1017 kg. The estimated amount of Kinetic Energy delivered by the impact is 1.3 x 1024 J. 1) Assuming the collision to have been perfectly inelastic, and that the recoil speed of the Earth be zero, calculate the possible range of velocities of the impactor and the relative linear momentum. 2) Assuming the duration of the impact to be 1 minute, what is the average force exerted by the impactor on the Earth? In both cases, explain your reasoning and show your calculations. 3) Supposedly something similar happened 4.5 billion years ago, when a Mars-size object hit planet Earth, penetrating all the way into its mantle and scattering material to a great distance. In time, due to the action of gravity, this material coalesced into our satellite, the Moon. Do you think that the reasoning you applied to the Chicxulub crater formation, applies to this case too? Why, or why not? Explain your reasoning.

Explanation / Answer

Solution:

1)The kinetic energy = 1/2 mv^2

1/2 mv1^2 = 1.3 e24 J

when m = 10^15 kg,

v= square root (2* 1.3 e24 / 10^15 )

v= 50990 m/s

when m = 4.6 e17 kg, v = Square root [2 (1.3 * 10^24)/(4.6 x 10^17) ] = 2377 m/s

the range of velocities =2377 m/s to 50,990 m/s

2) Force = change in momentum / time = energy / time * velocity

= [(1.3 e24 J )/ 60 s] * 50990 = 1.1 x 10^27 N

F = 1.3 e24 * 2377 / 60 = 5.15 x 10^27 N

3) It applies since the force of impact depends on the change in momentum and the duration of the ipact.

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