Consider a mobile moving along this path that starts at the point #(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

Answer:

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

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)#