Can someone explain the math behind snooker ball pocketing and power transfer?

[chickenguy]

Hi everyone, i am doing a maths project and i need some help on finding out about the maths of snooker. i.e.angles, power in relation to how the ball is pocketed and how power is transferred from one ball to another(this might be a bit hard for me to understand since i am only in year 8)

 

[ramollari]

It is not just Maths, it is Physics!
So what do you want to know about the physics of snooker?
It is all about collisions and conservation of momentum.
- The collisions between snooker balls are approximately elastic, so you should consider the conservation of energy (and of momentum).
- The hit ball continues to move along the line that joins the centers of the two collided balls, except if they are of unequal size (quite frequent).
- To know the speed you have to give to each ball, the rotational coefficient of friction \mu_{rot} is involved.
- The angles are just needed to break into components to apply the conservation of momentum, when the balls are of unequal size.

 

[maverick280857]

Hi Chickenguy

Nice to know your interest. Well, you do need a relatively higher level of mathematics to completely comprehend collisions. But let us start with a simple set of ideas.. What happens when two masses collide? They exchange energy and momentum. Momentum is the product of their mass and velocity.

I assume you have no experience with vectors. A vector is a quantity which has magnitude and direction. For instance a car traveling at 40km/hr west has a velocity of 40km/hr in the west direction. So velocity is a vector. On the other hand, the speed of the car is 40km/hr and has no direction. Statements like "speeding in that direction" refer to velocity actually.

Now in physics, certain quantities remain unchanged during an event. These quantities are said to be conserved and the laws which describe their conservation quantitatively are known as Conservation Laws. They are considered as powerful tools of physics. Momentum is one such quantity which is conserved during a collision under certain circumstances. This allows me to write the following equation,

Momentum before collision = Momentum after collision

To make the task of accounting and visualization easier we consider an aggregate of bodies as a SYSTEM. The momentum conservation equations for a system can be written as above and must include individual terms for each body before and after collision. Without resorting to much mathematics, you can easily see that speeds may be changed after collision of two snooker balls in such a way that the momenta remain the same before and after collision.

So far so good but why do you need so much mathematics to explain momentum in a better way? Well, this isn't all that is to collision you see. Collision involves several stages: collision, deformation, restitution, etc. Again an indepth analysis of all these stages is not normally carried out in general physics and the collision process is assumed to be ideal enough to neglect any major deviations from the laws we use. Hence, you may safely neglect these deviations for now (as you will study them much later and in a much better way when they will come naturally to you).

Essentially there are two kinds of collisions: elastic and inelastic. Kinetic Energy (= half the mass times square of speed) is conserved in case of an elastic collision and is not conserved in case of the more real inelastic collision (in case of snooker balls, inelastic collision takes place and that's how power is transferred). We should really say that energy is transferred but since power is the rate of change of energy with time, it does not make a lot of difference.

Coming back to your problem. You need to find out how Newton's Laws of Motion and Momentum Conservation principles assist you in setting up a physical model of collision of two balls first. Once you have the idea clearly set in your mind you can extend your model to include three, four or possibly more balls. As you are in grade 8, a physical picture is (I believe) a much nicer idea for you than diving into mathematics straight without backgrounds.

On the other hand if you have some time and enthusiasm to read some background material so that you can understand some of this mathematics then you might consider googling this up. Additionally you might want to check out a general physics textbook which does not use calculus. There is a college level book by Cutnell and Johnson which uses zero calculus and little algebra (as much as you would know) and has reasonably good physical descriptions for your purpose. I'll try and fish out some other names in the mean time.

If you have any more questions, please feel free to reply here and I'll be glad to help you out mate :-)

Cheers
Vivek

 

[maverick280857]

It is not just Maths, it is Physics!
So what do you want to know about the physics of snooker?
It is all about collisions and conservation of momentum.
- The collisions between snooker balls are approximately elastic, so you should consider the conservation of energy (and of momentum).
- The hit ball continues to move along the line that joins the centers of the two collided balls, except if they are of unequal size (quite frequent).
- To know the speed you have to give to each ball, the rotational coefficient of friction \mu_{rot} is involved.
- The angles are just needed to break into components to apply the conservation of momentum, when the balls are of unequal size.

Mate I doubt if he'll understand all the things you want to tell him because he's in grade 8 without calculus and conservation.

 

[ramollari]

OK, you did a good job in reducing it to pure math.

 

[chickenguy]

ok, you two havejust confused me pretty well so maybe i'll think about somrthing else, however, if you could get me any info on 0.999999999...=1, that would be great. also, i will need to conduct a survey, so can you tell me if you think that 0.99999999999.....=1? and also why you think so

 

[HallsofIvy]

A survey? Are you under the impression that whether a mathematics statement is true of false is subject to vote.

Anyone who knows what numbers are knows (not "thinks") that
0.9999... is just a different way of writing 1, just as 1/2 is just a different way of writing 0.5.

 

[chickenguy]

Will someone hurry up and answer already? my time for the project is running out