I recently re-watched a Sixty Symbols video in which the wonderful Ed Copeland describes the unexpected appearance of in a problem about bouncing balls.
The set-up is as follows: consider two masses and moving in one dimension on a smooth, horizontal surface. The bigger mass approaches the smaller mass at speed The two masses collide; the smaller mass recoils, bounces off a wall behind it, and moves back towards the bigger mass. The two collide again, and again, and again; every time, the bigger mass slows down a little as the smaller mass drums against it. Eventually the bigger mass will stop, having had all of its energy stolen by the smaller mass.
Here’s the interesting result: if the two masses are related by the equation
the number of collisions undergone by the two balls before the bigger mass changes direction spells out the first digits of .
So if the bigger mass is 16 times greater than the smaller mass, there will be 3 collisions before the bigger mass turns around; if the bigger mass is 1 600 times greater than the smaller mass, there will be 31 collisions, and so on.
The goal of this post is to provide a derivation of this result, relying heavily on the use of 2×2 matrices. I’ll do my best to omit all the algebraic manipulation that doesn’t need to be seen. I hope you’ll follow me all the way!
The first objective is to find an expression for the velocities of the two masses after their th collision.
We’ll first introduce some notation. The symbol represents the velocity of the body with mass immediately before the th collision, and the symbol is its velocity immediately after the th collision.
Solving this pair of simultaneous equations for and is a bit boring, so I’ll just quote the results:
We can summarise this information by introducing the ratio , defined by
and writing the simplified equations in matrix form:
Next, we have to relate the velocities before the th collision to the velocities after the th collision. The bigger mass’ velocity doesn’t change between collisions, so
but the smaller mass bounces (elastically) off a wall, so its velocity changes sign:
Again, we can write this in matrix form:
Substituting this expression into the other matrix equation above gives
By the rules of matrix multiplication, this is equivalent to
This recursive equation allows us to work out the velocities of the two masses after the th collision in terms of their initial velocities:
Finding the expression for a matrix raised to a power is in general difficult, unless we find a clever way of rewriting it.
Let’s give the matrix above a name:
Now suppose we can write the matrix in the form
where is some matrix, is its inverse, defined such that
and the filling of the sandwich is a diagonal matrix, one of the form
If we can write in this form, then
Since is a diagonal matrix, the evaluation of is not complicated by funny cross-terms:
This would allow us to find an explicit expression for . This technique is known as matrix diagonalisation.
But how do we find expressions for both and ? Since
we can let act on the right-hand side, giving
Let’s write in the form
where the are two-component column vectors. Then
By equating the components it must be true that
The vectors that satisfy this equation are said to be the eigenvectors of , and the are their corresponding eigenvalues. So the matrix is the matrix whose columns are the eigenvectors of , and is the diagonal matrix whose non-zero elements are the eigenvalues of .
How do we find the eigenvalues and eigenvectors of a matrix? We could always guess them simultaneously, but this is difficult in general. Fortunately there’s an algorithm we can use to work out the eigenvalues of a matrix, using the following reasoning: the eigenvalues of satisfy the equation
We are free to insert an identity matrix into the right-hand side:
This equation can in principle be solved by applying the inverse of to both sides, giving
but this is not the case we’re interested in; we want non-zero eigenvectors! If the inverse of the matrix exists, we will be stuck with this solution. We therefore conclude that for , the inverse of does not exist.
If a matrix has no inverse, its determinant, defined through
is equal to zero. So the eigenvalues of satisfy the equation
This is the so-called characteristic equation. Its solutions are
We’ll label these two solutions with a plus or minus subscript, such that
This gives us the elements of the diagonal matrix . We also need the eigenvectors that correspond to these eigenvalues, which by definition satisfy
We find the two eigenvectors are
Hence our two matrices and are
The last bit of work to do is to find the inverse of the matrix . I’ll just write down the answer:
Let’s substitute these results into our expression for the velocities of the masses:
After lots of matrix multiplication, we are left with the result
We’re very close! We just need to simplify this equation using the expressions
Raising a complex number to a power in Cartesian form is not easy, so we’ll convert to polar form instead:
Substituting this expression into that for gives
Now the awesome bit:
So the velocity of the larger mass varies with the cosine of the number of collisions it has suffered. Nice!
Now we ask the question: after how many collisions is the velocity of the greater mass negative? That is, after how many collisions is the bigger mass guaranteed to have changed direction? Since cosine goes to 0 when its argument reaches , must satisfy the equation
Take the special relation quoted in the video:
For any ,
so we can write
The last step: we can use the Taylor expansion of the arctangent function to say that
If satisfies this inequality, the velocity of the larger mass is negative, ie it has collided with the smaller mass sufficiently many times that its direction has been reversed. For example, if ,
This means the bigger mass acquires a negative velocity at =4, so the number of collisions that occur before this happens is 3.
Let’s try a bigger number. If ,
This means the bigger mass acquires a negative velocity at =315, so the number of collisions that occur before this happens is 314.
Hence in general, the number of collisions that occur before the one which causes the big mass to change direction is the greatest integer that fails to satisfy the inequality above, written
where the brackets represent the floor function.
Here’re some animations simulating the encounter. This first animation shows the case where the bigger mass is 16 times greater than the smaller mass.
Here’s the case where the bigger mass is 1 600 times greater than the smaller mass.
The larger mass is much harder to turn around this time. This is because the big mass can only impart a finite amount of energy to the smaller mass each collision, so more collisions are needed to stop the big mass. Notice how rapidly the small mass moves when the large mass is stationary – all of the kinetic energy initially possessed by the big mass is concentrated into the small mass. Since the (classical) kinetic energy of an object is given by
and the large mass is 1 600 times heavier than the small mass, the small mass must move 40 times faster than the big mass does initially if energy is conserved! This is faster than the screen capture software can keep up with, which is why the small mass’ motion looks a little strange in the animation.
As predicted, there are 31 collisions before the big mass is made to turn around.
Unfortunately, this derivation doesn’t provide an intuition as to why should appear. It’s worth emphasising that it has nothing to do with the shape of the two masses, which were pictured as circular by accident.
The appearance of stems from the appearance of the cosine function; the appearance of the cosine stems from the symmetry of the eigenvalues of ; the symmetry of ‘s eigenvalues stems from its own symmetry; and the symmetry of the matrix stems from the conservation of momentum and energy. But this doesn’t answer the question “where is the hidden circle that’s making appear?”. Here’s an interesting post which addresses this question.
Thanks for reading!