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# Number Theory/how to determine angle of reflection, after n reflection inside a square

Question

reflection inside a sq
Can we determine at what angle, a line will reflect, after hitting n
times the sides of a square.

Lets say we draw a line at a certain angle on one side of a square. It
then reflect to the other side of the wall at a certain angle. And it
continues reflecting on the side of the wall of  the square n times.

Is there a general formula that can tell at what angle it will make,
after reflecting n times on the side of a square.

There is certainly a formula that can tell, given the angle it make
first with a wall and at what angle it will make on the other side of
the square, after reflecting. But what angle it will make after
reflecting n times, is there any?

Is it right, we have to go calculating for each reflection to get the
angle, after n reflection?

Isn't, this is how a computer find the angle after n reflection?

One thing is for sure, for a certain square, for each angle it make at
a certain point on one side of a square, after n reflection on the
square, it will hit a certain side at a certain point and at a certain
angle.

But I just can't think of a general formula, that can tell at what
point or at what angle it is going to hit on a certain side of the
square.

Thank you.

If the initial angle made with the side of the square is x, and the coefficient of restitution is unity, after an odd number of bounces, the angle will be 90-x, and after an even number of bounces it will be x.
If the coefficient of restitution is e, after an odd number of bounces, n, the angle will be given by
arctan(e^n*cot(x)) and after an even number of bounces arctan(e^n*tan(x)).  This means that as time goes on, the ball gets closer to the sides of the square.

Regards

vijilant
Questioner's Rating
 Rating(1-10) Knowledgeability = 10 Clarity of Response = 10 Politeness = 10 Comment Thanks a lot, the reply was so soon. I didn't understand coefficient of restitution but I will have a look at it.

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Number Theory

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#### Vijilant

##### Expertise

Most questions on number theory, divisibility, primes, Euclidean algorithm, Fermat`s theorem, Wilson`s theorem, factorisation, euclidean algorithm, diophantine equations, Chinese remainder theorem, group theory, congruences, continued fractions.

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Teacher of math for 53 years

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AQA Doncaster Bridge Club Danum Strings Orchestra Doncaster Conservative Club Danum Strings Orchestra Simply Voices Choir Doncaster TNS mystery shopping St Paul's Music Group Cantley

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Journal of mathematics and its applications M500 magazine

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BSc (Hons) Liverpool (Science). BA (Hons) OU (Mathematics)

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State Scholarship 1955 Highest Score in Yorkshire on OU course MST209 £50 prize First class honours in OU BA Mathematics

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I taught John Birt, former Director of the BBC in 1961. His homework book was the most perfect I have ever marked. And also the most neat. I could tell he was destined for great things. One of my classmates was the poet Roger McGough, and I have a mention in his autobiography.