page_title: Gravity train
section_title: Straight path between two arbitrary points
hierarchical_section_title: Gravity train / Mathematical derivation / Straight path between two arbitrary points
caption_attribution_description: English: Diagram of forces on a gravity train on non-diametrical straight line path.
context_page_description: A gravity train is a theoretical means of transportation for purposes of commuting between two points on the surface of a sphere, by following a straight tunnel connecting the two points through the interior of the sphere.
  In a large body such as a planet, this train could be left to accelerate using just the force of gravity, since during the first half of the trip, the downward pull towards the center of gravity would pull it towards the destination. During the second half of the trip, the acceleration would be in the opposite direction relative to the trajectory, but, ignoring the effects of friction, the speed acquired before would be exactly enough to overcome this deceleration, and as a result, the train's speed would reach zero at precisely the moment the train reached its destination.
context_section_description: For the more general case of the straight line path between any two points on the surface of a sphere we calculate the acceleration of the body as it moves frictionlessly along its straight path.
  The body travels along AOB, O being the midpoint of the path, and the closest point to the center of the Earth on this path.  At distance  along this path, the force of gravity depends on distance  to the center of the Earth as above.  Using the shorthand  for length OC:
  The resulting acceleration on the body, because is it on a frictionless
  inclined surface, is :
  But  is , so substituting:
  which is exactly the same for this new , distance along AOB away from O, as for the  in the diametric case along ACD.  So the remaining analysis is the same, accommodating the initial condition that the maximal  is  the complete equation of motion is
  The time constant  is the same as in the diametric case so the journey time is still 42 minutes; it's just that all the distances and speeds are scaled by the constant .
caption: Diagram of forces on a gravity train on non-diametrical straight line path
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