Consider a mobile moving along this path that starts at the point $(0, 0)$ $(0,0)$ . What are the coordinates of the point of arrival of the mobile ? (See image below)

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A "Suits and Series" problem.

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Answer 1 · 2018-07-30T21:46:20

Point of arrival is: $(\frac{32}{7}, \frac{10}{3})$ $(32/7,10/3)$

In the x-direction, defined as horizontal direction with positive direction being left to right:

$x = 8 - 6 + 9/2 - 27/8 + ...$

This is an alternating geometric series.

$= 8 - 8(3/4) + 8(3/4)^2 - 8(3/4)^3 + ...$

$= 8 ( underbrace(( 1 + (3/4)^2 + ... ))_(a = 1 qquad r = (3/4)^2) - underbrace(( 3/4 + 3/4 (3/4)^2 + ...))_(a = 3/4 qquad r = (3/4)^2 ) )$

The sum to infinity for a geometric series with $abs r lt 1$ is:

$""_xS_oo= 8 ( 1/(1- (3/4)^2 ) - (3/4)/(1- (3/4)^2 ))$

$:. ""_xS_oo= 32/7 qquad qquad qquad [= 3.375]$

In the y-direction, defined as vertical direction with upward positive:

$y = 6 - 25/4 + 96/25 - 384/125 + ...$

$= 6 - 6(4/5) + 6(4/5)^2 - 6(4/5)^3 + ...$

$= 6 ( (1 + (4/5)^2 + ... ) - ( (4/5) - (4/5)^3 - ...) )$

With the same reasoning:

$""_yS_oo= 6 ( 1/(1- (4/5)^2 ) - (4/5)/(1- (4/5)^2 )) = 10/3$

Point of arrival is: $(32/7,10/3)$