Consider three axes of rotation for a pencil: along the lead, at right angles to the lead at the middle, and at right angles to
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The total average velocity 
(I assume west as positive direction) is given by the total displacement, S, divided by the total time taken, t:
where:
-The total displacement S is the algebraic sum of the displacement in the first part of the motion (
, due west) and of the displacement in the second part of the motion (
, due east).
-The total time taken t is the time taken for the first part of the motion,
, and the time taken for the second part of the motion, 
. can be found by using the average velocity and the displacement of the first part:
, instead, can be written as 
, where 
is the average velocity of the second part of the motion (with a negative sign, since it is due east).
Therefore, we can rewrite the initial equation as:
And by solving it, we find the displacement in the second part of the motion (i.e. how far did the backpacker move east):
where:
-The total displacement S is the algebraic sum of the displacement in the first part of the motion (
-The total time taken t is the time taken for the first part of the motion,
Therefore, we can rewrite the initial equation as:
And by solving it, we find the displacement in the second part of the motion (i.e. how far did the backpacker move east):