Question

in an experiment, a
bump car is allowed to travel freely from one end to another in a meter stick. When it hits one end, the car bounces back 6 meters and then move forward again. On each bounce, it moves back 2/3 of the distance is it previously , traveled. a.) determine the distance covered by the bump car from the head wall after four bounces b.) solve for the total distance traveled by the car after 5 bounces
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Answer 1

Given : When it hits one end, the car bounces back 6 meters and then move forward again.

On each bounce, it moves back 2/3 of the distance is it previously , traveled.

To Find : a.) determine the distance covered by the bump car from the head wall after four bounces

b.) solve for the total distance traveled by the car after 5 bounces

Solution:

Distance covered after 1st bounce = 6 m Backward and then 6 m forward

= 6 + 6 = 12 m

Distance covered after 2nd bounce = (2/3)6 = 4 m  Backward and then 6 m forward

= 4 + 4  = 8 m

Distance covered after 3rd bounce = (2/3)4 =8/3 m  Backward and then 8/3 m forward

=8/3 + 8/3  = 16/3 m

Distance covered after 4th bounce = (2/3)(8/3) =16/9 m  Backward and then 16/9 m forward

=16/9 + 16/9  = 32/9 m  = 3.55 m

Distance covered after 5th bounce = (2/3)(16/9) =32/27 m  Backward and then 32/27 m forward

=32/27 + =32/27  = 64/27 m

total distance traveled by the car after 5 bounces

= 12 + 8  + 16/3  + 32/9  + 64/27

This is a GP  a  = 12  , r = 2/3 , n = 5

Sₙ = a( 1- rⁿ)/(1 - r)  = 12 (  1 - (2/3)⁵)/(1 - 2/3)  

= 12 ( 1 - 32/243)/(1/3)

= 12(211)/81

= 4(211)/27

= 844/27

=  31.259

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