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.

1 Answer
Jul 30, 2018

Point of arrival is: (32/7,10/3)(327,103)

Explanation:

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:

  • S_oo = a/(1 - r)

""_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)