Can conservation principles solve the killer asteroid problem?

[jahlex]

The problem:

Killer Asteroid
Some data:
Mass of Earth = 6.0 x 10²⁴ kg
Radius of Earth = 6.4 x 10³ km
Gravitational constant G = 6.67 x 10⁻¹¹N-m/kg²

You receive an urgent phone call from NASA. Their scientists have just detected an asteroid headed for Earth. Its mass is 6 x 10 15 kg, which is only one billionth the mass of Earth, but even so, larger than the dinosaur extinction asteroid. The asteroid is 4.5 x 10⁹km away from Earth’s center and is moving at 2.5 x 10⁴m/s. The asteroid’s present heading gives it a straight-line trajectory (i.e. its path if there was no gravity) that would miss Earth’s surface by only 3200 km. Quick! Will the asteroid strike Earth?

My problem with this problem:

What my professor started to do on the board was ignored the v₀ = 2.5 x 10⁴m/s parameter provided and said there was some initial velocity v you could solve for such that the asteroid would just graze the surface of the Earth tangentially, and if v₀ < v then the asteroid would collide with Earth but if v₀ > v then the asteroid would fly into space. At this point I got a nosebleed and had to leave the classroom before seeing how he solved for v on the board.

I think we were supposed to use the three conservation principles to solve this type of problem (cons. of linear momentum, cons. of angular momentum, cons. of mechanical energy) but I don't see how it's possible to use the first two principles because there is a net force and there is a net torque (With the center of Earth as the origin). If you only set up a relationship with conservation of mechanical energy, then you can solve for a velocity of the asteroid at the surface of the Earth from the initial kinetic energy and the difference in potential energy, but there is no way to tell whether the asteroid would be grazing the surface or striking the surface.

I'm not really seeking a solution so much as principles by which to approach this species of problem because it seems that the conservation principles aren't enough. Thank you in advance.

 

[Dick]

It sounds like you are ignoring the rest of the solar system. That's fine, if unrealistic. So you just have the asteroid and the earth. Since the asteroid is so small relative to the mass of the Earth you can assume the Earth is stationary. Relative to the center of the Earth the acceleration of the asteroid is directed to the center of the Earth and so is the radius vector. So, in fact, there is approximately no torque on the asteroid. Angular momentum of the asteroid is conserved. Use that.