In the previous post in this series I produced a table of acceleration due to gravity against depth from surface, based on the four layers of the Earth (Crust, Mantle, Outer Core, Inner Core), assuming a constant density for each layer.  This showed that the variation of acceleration is far from linear, and that the maximum acceleration occurs not at the surface, but at the boundary between the Mantle and Outer Core. 

This table of accelerations can now be used as input to the ODE solver function to give a more accurate estimate of the time taken for a ball dropped down the Hole Through the Centre of the Earth to reach the centre (keeping all the other simplifying assumptions, such as perfect vacuum, no impact with the sides, etc).  The spreadsheet containing this calculation can be downloaded from ODESolver-Buckle.zip 

The revised function to be solved is shown in the screen shot below: 

Revised function for the ball through the centre of the Earth problem

Note that the CoeffA argument now contains an array of acceleration values at selected depths, rather than a single rate of change of acceleration value.  The CSpline function is used to return the slope of a cubic spline passing through these points, at the distance from the centre specified in Y(0). 

Input and part of the output results are shown below: 

ODE function input and output

The time to reach the centre of the Earth, taking account of the actual variation of acceleration with depth, has reduced from 1264.5 seconds to 1168.5 seconds (19 minutes 28.5 seconds), a reduction of about 7.6%. 

Graphs of distance from the centre, velocity, and acceleration against time are shown below, with the corresponding graph assuming linear variation of acceleration.  It can be seen that the maximum velocity has increased from 7,897 m/s to 9,636 m/s, and the acceleration at the Mantle/Outer Core boundary has increased from 5.4 m/s² to 10.32 m/s². 

Distance from Centre vs Time

Velocity vs Time

Acceleration vs Time