Quandaries and Queries
  

 

Name: Eman
Who is asking: Student
Level: Secondary

Question: Hi there ~

I am doing my AS levels and have come across something that I am stuck with. I planned to ignore it but it keeps coming up literally everywhere, so I guess its important to know how to do it. I’d really appreciate it if you would please *explain* HOW to do it, because I have the solution to one of the sums of the type, but its beyond my understanding as to what exactly I am to do and WHY.

Here it is:

Q : When a child’s ball is dropped from a height h metres on to a hard, flat floor, it rebounds to a height of 3/5h metres. The ball is dropped initially from a height of 1.2m.

  1. Find the maximum height to which the ball rises after two bounces.
  2. Find the total distance that the ball has traveled when it hits the floor for the tenth time.
  3. Assuming that the ball continues to bounce in the same way indefinitely, find the total distance that the ball travels.

Thank you so much for helping. You guys are doing a great job. :) I am doing this course through correspondence so would really appreciate the help.

Thanks again.

Eman.

 

 

Hi Eman,

If the ball starts at height h meters then in one bounce it goes to the floor and then bonces up  3/5h meters for a total distance travelled of

h + 3/5h = 8/5h meters For the second bounce the ball is at height  3/5h meters and hence travels to the floor (a distance of  3/5h meters) and then bounces back to  3/5( 3/5h) meters for a distance travelled on the second bounce of  3/5h + 3/5( 3/5h) =  3/5(h + 3/5h) =  3/5( 8/5h) meters Hence, after 2 bounces, the total distance travelled is  8/5h +  3/5( 8/5h) =  8/5h(1 +  3/5) meters In the same fashion you can see that after 3 bounces the ball has travelled  8/5h(1 +  3/5 + ( 3/5) 2) meters

After 9 bounces the ball has travelled

 8/5h(1 +  3/5 + ( 3/5) 2 + ... + ( 3/5) 8) meters and is at a height of ( 3/5) 9h meters. Hence when it hits the floor the tenth time it has travelled a distance of  8/5h(1 +  3/5 + ( 3/5) 2 + ... + ( 3/5) 8) + ( 3/5) 9h meters

For part 3, you have  8/5h meters times a geometric series

1 +  3/5 + ( 3/5) 2 + ... + ( 3/5) n + ... To see how to sum this series look at Infinite Geometric Series. Using the notation in this note you have a = 1 and r = 3/5. Since r is positive and less than 1 you problem is cover by case 1.

I hope this helps,
Penny 		 
 

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