A neutron star has radius $10$km and mass $2.5\times 10^{29}$kg. A meteorite is drawn into its gravitational field. Calculate the speed with which it will strike the surface of the star. Neglect the initial speed of the meteorite.
In the solution provided, we use the formula for escape speed, which is
$$v = \sqrt{\frac{2GM}{R}}$$
But the derivation of the formula for escape speed involves a mass has zero kinetic energy and zero gravitational potential energy.
So, my question is: Why can we use escape speed to calculate striking of a meteor in this case?
Best Answer
This is a classic example of the conservation of total energy. The meteor is assumed to have zero kinetic energy sufficiently far away from the black hole and the gravitational potential there is zero, as well. As the meteor gains kinetic energy while falling toward the black hole the gravitational energy increases in direct proportion (though the latter is negative so the sum does not change). When reaching the surface of the black hole the meteor's kinetic energy is equal to the potential energy there, as well.
$$1/2mv^2 = MG/R $$
Turning the problem around, you have the (escape) velocity required to get the meteor unbound from the black hole's gravitational potential.