Remove multiples and sum a bouncing path

(a) Find the sum of all integers in {1,2,3,…,2014}\{1,2,3,\ldots,2014\} that are not multiples of 3. marks

(b) An elastic ball released from height hh falls vertically onto a horizontal floor and rebounds to height ahah, 0<a<10<a<1. Initially it is released from height HH.

(i) Write an expression in H,a,nH,a,n for its height after bounce nn. mark

Given a=34a=\frac34,

(ii) find the number of bounces before the height is less than H/20H/20 for the first time marks;

(iii) calculate in terms of HH the total distance just before it comes to complete rest. marks

[2014/NYJC/JC2/mid-year/P2/4]

Total marks: 10

Solution

(a)

Exclude the multiples of 3

Sum the integers from 1 to 2014 that are not multiples of 3.

Hint

  1. Find the largest multiple of 3 that does not exceed 2014.

Answer · (a)

13527371352737.

(b)(i)

Height after n bounces

Find the rebound height after bounce nn.

Hint

  1. Check your formula at n=1.

Answer · (b)(i)

anHa^nH.

(b)(ii)

First rebound below H/20

For a=3/4a=3/4, find the first bounce with rebound height below H/20H/20.

Explore this idea

Hint

  1. Cancel H before taking logarithms.

Answer · (b)(ii)

Bounce 1111.

Explore the idea

Rebound height as a fraction of H

The initial release is n = 0. Each bounce multiplies the height by 3/4.

Bounce countn = 11
Try an example
hₙ / H at n = 11
0.0422
Key idea

Bounce 10 is still above H/20; bounce 11 is below.

Ideas to try
Compare the values on either side of the boundary. What makes one count qualify?
(b)(iii)

Total distance travelled

For a=3/4a=3/4, find the total distance travelled before rest.

Hint

  1. Separate the first fall from the up-and-down rebound trips.

Answer · (b)(iii)

7H7H.

Takeaway

Count each journey once before summing

Every rebound height is travelled upward and downward; the initial drop is travelled only once.