Question

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)

A "Suits and Series" problem.

Answer 1

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)`