# Asteroid Problem

On Wednesday 9th of November 2011 asteroid 2005 YU55 passed by the Earth at a distance of 323,469 km, which is closer than the Moon. There was no possibility of it hitting the Earth - this time. Just for interests sake, assume there was an asteroid heading for Earth, and it was necessary to blow it up, so that instead of a large chunk of rock potentially ploughing into a densely populated city we ended up with a spectacular meteor shower as the fragments burned up in the outer atmosphere. Also assume an astronaut lands on the asteroid to plant the explosives and then just using the strength in their legs "jumps" off again to return to their spaceship. What is the maximum radius the asteroid can have for the astronaut to be able to jump completely away from it assuming the average density of the asteroid is the same as the Earth? The astronaut, in full spacesuit can jump about half a metre on Earth.

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On Wednesday 9th of November 2011 asteroid 2005 YU55 passed by the Earth at a distance of 323,469 km, which is closer than the Moon. There was no possibility of it hitting the Earth - this time. Just for interests sake, assume there was an asteroid heading for Earth, and it was necessary to blow it up, so that instead of a large chunk of rock potentially ploughing into a densely populated city we ended up with a spectacular meteor shower as the fragments burned up in the outer atmosphere. Also assume an astronaut lands on the asteroid to plant the explosives and then just using the strength in their legs "jumps" off again to return to their spaceship. What is the maximum radius the asteroid can have for the astronaut to be able to jump completely away from it assuming the average density of the asteroid is the same as the Earth? The astronaut, in full spacesuit can jump about half a metre on Earth.

The question is related to the concept of escape velocity which is the velocity of an object required to through it out of the gravitational field of a planet.

For a planet escape velocity is given by

Here G is the universal gravitation constant, M is the mass of the planet and R is its radius.

If the density of the planet is r then above formula can be written as

-------------------------- (1)

Now as the astronaut can jump up to a height h of half meter the maximum velocity it gains by jumping is given by (conservation of energy)

Or ----------------------------------------- (2)

As we know that the acceleration due to gravity at the surface of earth is

(Density of asteroid is same as earth material)

Substituting in (2) we get

Now to jump completely away from the asteroid the velocity of jumping must be equal to the escape velocity at the asteroid, thus

Or

Or

Or

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