Billiard Ball Problem

Billiard ball problem can be seen as one elegant mathematical problem. In this article, we will analyse it mathematically.

Before we begin, let us first see the problem. Given below is the situation. Can you predict, in which pocket will the ball falls in?

Let us now connect this problem with mathematical conditioning. We all know reflection property very well. Now what you have thought is that whenever ball strikes any mirror then it enters into a rectangle which is just a mirror image of it. Now just keep on constructing rectangles with taking common sides CD and BC (initially) then their reflections too.

Remember one thing that the ball will keep on moving in its direction until it reaches one of the vertices of our apparent rectangles. and hence it is obvious that the no. of such apparent rectangles that you will need in the horizontal direction will be 67 and in the vertical direction is 100 (smallest possible). being 67 odd hence, it is obvious that it will fall either in C or D and 100 being even it will eventually fall into D.

Alternatively, we can think, as to travel from the lower-left pocket to any other pocket, the ball must traverse integer multiples of the length and width (that is, the ball must travel so that it is either at the top/bottom and right/left of the pool table to land into a pocket). We also know that since it is launched at a 45 degrees angle its speed is split evenly into the horizontal and vertical component (this means at any point it will have travelled the same horizontal and vertical distance).

If we let the horizontal lengths travelled by “x”, and the vertical widths travelled be “y”, then:

100x = 67y where x,y are integers

And, the first value (s) that this works for is x=67 and y=100. This means the ball will have travelled an odd number of lengths (ends up at the right) and an even number of widths (ends up on the bottom), and we can easily conclude from this that it ends up in the bottom right pocket, giving the answer of D.

Wait think about it again! What all we had done so far is that we have computed the lowest common multiple of the length and width, for a problem that seemed so much more complicated than that. Is not it Mind-blowing!

Comments

Time Management: Pomodoro Technique

Are you facing challenges in managing your time properly? If you often waste time on unproductive tasks and finally at the end of the day feel you have wasted your day. Pomodoro Technique may be a good solution to all your problems. Today I am going to discuss here, a very useful technique to improve your productivity and better time management. The credit for this technique goes to Francesco Cirillo. He utilized a tomato-shaped sandglass to manage his time effectively. He called it the Pomodoro technique (which is an Italian word for “tomato”). The idea behind the Pomodoro Technique is to break down all of your tasks into 25-minute time blocks. Between two session , keep a five-minute break. And after completing four Pomodoros sessions taking an extended break usually 15-20 min. There are six steps in the original technique: Decide on the task to be done. Set the Pomodoro timer (traditio...

Ramanujan in London

When super genius Indian Mathematician Srinivasan Ramanujan arrived London, he was greeted by Professor Godfrey Harold Hardy. Just to break the silence , Hardy commented on the Taxi number , he came in is 1729- " seems like an uneventful number". Ramanujan had a cursory glance at the taxi number plate himself and replied casually "Oh No, actually it's a really interesting number. It is the smallest number representable in two alternative ways as the sum of two cubes and then this brilliant man told the equation on the spot. 1729 is the sum of the cubes of 10 and 9 - cube of 10 is 1000 and cube of 9 is 729; adding the two numbers results in 1729. 1729 is also the sum of the cubes of 12 and 1- cube of 12 is 1728 and cube of 1 is 1; adding the two results in 1729.

IIT Delhi Launches IITPAL Portal

The Indian Institute of Technology (IIT) Delhi launched an interactive IIT- PAL website —iitpal.iitd.ac.in — to give free videotape lectures to class 11 and 12 scholars who are preparing for competitive examinations similar as JEE, NEET, IAT and others. www.iitpal.com This action of the Ministry of Education (MoE) had started with an end to make their understanding of the wisdom subjects (Physics, Chemistry, Maths and Biology) better and to help tone- studying scholars do well in competitive examinations like JEE, NEET, IAT and others. Now, to bring all coffers together, IIT-D has launched a website. The website —iitpal.iitd.ac.in — will act as a single platform where scholars across India can pierce videotape lectures that are telecast on the Ministry of Education’s Swayam Prabha Channels, interact live with IIT Professors and ask them questions. “ This website will be helpful to scholars especially from regions where they may not have access to specialist prec...

IITPAL

Search This Blog