gravity - What would happen if a body were to fall into a neutron star?

Let's assume that what is falling onto the neutron star is "normal" material - i.e. a planet, an asteroid or something like that. As the material heads towards the neutron star it gains an enormous amount of kinetic energy. If we assume it starts from infinity, then the energy gained (and turned into kinetic energy) is approximately (ignoring GR)
$$frac{1}{2}m v^2 = frac{GMm}{R},$$
where $m$ is the mass of the object (which cancels) and $M$ and $R$ are the mass and radius of the neutron star (let's assume typical values of $1.4 M_{odot}$ and 10 km respectively).

This results in a velocity as it approaches the neutron star surface of $1.9 times 10^{8}$ m/s - i.e. big enough that you would have to do the calculation using relativistic mechanics actually.

However, I doubt that the object would get to the surface intact, due to tidal forces. The Roche limit for the breakup of a rigid object occurs when the
object is a distance
$$d = 1.26 R left(frac{rho_{NS}}{rho_O}right)^{1/3},$$
where $rho_{NS}$ and $rho_O$ are the average densities of our neutron star and object respectively. For rocky material, $rho_O simeq 5000$ kg/m$^{3}$. For our fiducial neutron star $rho_{NS} simeq 7times10^{17}$ kg/m$^{3}$. Thus when the object gets closer than $d= 500,000$ km it will disintegrate into its constituent atoms.

It will thus arrive in the vicinity of the neutron star as an extremely hot, ionised gas. But if the material has even the slightest angular momentum it could not fall directly onto the neutron star surface without first shedding that angular momentum. It will therefore form (or join) an accretion disk. As angular momentum is transported outwards, material can move inwards until it is hooked onto the neutron star magnetic field and makes its final journey onto the neutron surface, probably passing through an accretion shock as it gets close to the magnetic pole, if the object is already accreting strongly. Roughly a few percent of the rest mass energy is converted into kinetic energy and then heat which is partly deposited in the neutron star crust along with matter (nuclei and electrons) and partly radiated away.

At the high densities in the outer crust the raw material (certainly if it contains many protons) will be burned in rapid nuclear reactions. If enough material is accreted in a short time this can lead to a runaway thermonuclear burst until all the light elements have been consumed. Subsequent electron captures make the material more and more neutron rich until it settles down to the equilibrium composition of the crust, which consists of neutron-rich nuclei and ultra-relativistically degenerate electrons (no free neutrons).

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