Acceleration from Gravity

Calvin and I were wondering how fast an object could go if it was pulled into a planet by gravity if it started a long ways away and there was not atmosphere to slow it down.

I did the calculation by calculating the gravitational energy at each distance away from the planet/star, then integrating to get the total energy after accelerating through that distance, and then converting that energy to velocity using the equation for kinetic energy.

Here's the plot for acceleration toward Earth:

So for earth, the maximum speed is about 25,000 mph which is only about 0.004% the speed of light.  (Graph generated with Wolfram Alpha using: y=sqrt(2*(9.8-9.8/x)*6.4e6)*2.237 from x = 1 to 100 and y = 0 to 25000 [9.8 is acceleration at surface of earth, 6.4e6 is radius of earth in m, 2.237 is conversion from m/s to mph]).

So what about the heaviest known object in the universe?  If we redo the calculation with the mass of the black hole = 4.2e40 kg (vs. 6e24 kg for earth) and the radius of the black hole of 5e13 m (vs. 6.4e6 m for earth) we get the plot below:

In this case, the object is calculated to reach a maximum 750,000,000 mph which is 10% faster than the speed of light! These calculations assume Newtonian physics and these break down when things approach the speed of light. But it's interesting that gravity alone can accelerate something close to the speed of light.