## Particle Collisions

Hi All,

It has been a long break on this blog. I was heavily busyb with some work. Anyways, lets get down to the business.

We are hearing buzz about CERN and the mightiest particle accelerator it created on which the “big bang” experiment has been started. And it so happened that I came across this nice puzzle, thanks to Sagarmoy.

Problem : 4 particles are travelling with mutually different uniform velocity ( uniform velocity implies that the speed is constant as well as the direction of motion does not change ).

We will say collision occurred between two particle when two particles are at the same place at the same time. Our particles being of a special kind, do not change their velocity even after the collision. You can imagine that particle can pass through the other.

The velocity vectors of these 4 particles are mutually non-parallel. Also, it is given that only 2 particles will participate in any collision ( no three line co-incide at the same point ).

Given these facts, we know that minimum 0 (no two lines are on the same plane) and maximum 6 collisions (${4 \choose 2}$) can occur. If it is known that 5 collisions have occurred, can you prove that 6th collision will always occur?

Solution : Well, there are two solutions. One by Sagarmoy and one by Deepanjan. I will describe Deepanjan’s solution first.

1) As shown in the figure, lines 1,2,3,4 indicate the direction of particle motions. Let’s say we want to know that when 5 collisions have occurred ( all but (3,4) ) then collision between (3,4) will always occur.

Let’s have a line 4′ parallel to line 4. We know that collision (1,4) and (1,3) has occurred. As line 4′ is parallel to line 4, it is easy to see that collision (1,4′) would also occur because the ration of distance needed to be travelled within a given timeframe for both the particles does not change. We also know that (1,3) collision has occurred. So obviously (3,4′) collision would also occur. Again, as 4′ is parallel to 4, (3,4) collision would definitely occur.

Similar proof can be given for any different orientation of lines.

2) This solution is by Sagarmoy and it quite elegant, I must say. Consider the path of particle in space-time ( 4-dimensions , 3 physical + 1 time). As the particles travel with uniform velocity, their path corresponds to lines in space-time. Again, let us say, we want to prove that collision (3,4) will occur if all other 5 collisions has occurred.

We know that collisions (1,2)(1,3)(2,3) has occurred. So, line 3 must lie on the plane defined by line 1 and line 2. Similarly we know that (1,2)(1,4)(2,4) has occurred, so line 4 also must lie on the plane defined by line 1 and 2. So, line 3 and 4 are on the same plane and are not parallel. It must happen that lines intersect. But intersection of lines in space-time means that particles are at the same place at the same time, so collision (3,4) must occur.

— Saurabh Joshi