Shaft

12-17-2007, 05:02 PM

Here is an attempt to settle, once and for all, the question of whether a light cue stick "breaks better" than a heavy cue stick.

The mathematical result surprised me, and it forces me to recant what I have said on previous posts that "it should not make a difference." It does, in fact, make a (small?) difference.

I apologize in advance for the use of metric units, but it makes the math so much easier. I will give the english equivalents wherever they help.

For the purpose of this analysis I will assume that

* The collision is perfectly elastic (energy and momentum will be conserved in the post collision motion of the stick and ball).

* The same shooter exerts a constant force (300 Newtons, about 67.4 lbs) over a 6 inch forward swing to impact. (The actual force and swing distance are not important here so long as they are the same for both cues.)

* The "light" stick weighs 18oz, the "heavy" stick weighs 21oz, and the cue ball weighs 6 oz.

* The stick and ball are free bodies in space. The the stick is accellerated to hit a ball that is initially at rest, and the axis and path of the stick is alligned with the center of the ball. (I know real balls and sticks are not free bodies in space, but this assumption simplifies everything without changing the outcome.)

The trick is to find the post collision speeds of the ball and stick so that energy and momentum are conserved. We have a system of two equations and two unknowns, so it can be solved with some hairy algebra. I used the Excel Solver routine because it was easier and I am LAZY.

A constant force of 300N applied over 6 inches will accelerate an 18oz (0.5103kg) stick to a speed of 13.38614 m/s (29.94mph). The pre-collision momentum of the stick is 6.830947 kg-m/s, and the kinetic energy is 45.72J. Conserving energy and momentum, the post collision speed of the stick is 6.69307m/s (14.97mph) and the post collision speed of the ball is 20.07m/s (44.92mph).

A constant force of 300N applied over 6 inches will accellerate a 21oz (0.5953kg) stick to a speed of 12.39368m/s (27.72mph). The pre-collision momentum of the stick is 7.377956 kg-m/s, and the kinetic energy is 45.72 J. Conserving energy and momentum, the post collision speed of the stick is 6.885376m/s (15.40mph) and the post collision speed of the ball is 19.27905m/s (43.13mph).

DRUM ROLL and TRUMPET FANFARE:::::::::::::

The cue ball has 4.1% more speed and 8.5% more energy when struck by a LIGHTER (18oz) stick than when it is struck by a HEAVIER (21oz) stick, all other things being equal. A 15% reduction in weight results in 4% more speed.

This is true - NOT because the lighter stick moves faster - but because the lighter stick transfers more energy to an object that is more similar in mass.

In fact, I demonstrated with the same spreadsheet that, as the mass of the two colliding objects gets more and more equal, more and more of the energy of the first object is transfered to the second object. When the mass difference is zero, the energy transfer is 100%.

You might find that hard to believe. Remember those annoying clickety-clackety novelty toys that had 5 steel ball bearings suspended on a wooden frame? Take three balls out of the way and try it with just two balls. The first ball will stop dead as the second ball takes 100% of the energy and swings upward.

We can also see this principle on the pool table. A dead-stratight stun shot transfers 100% of the cue ball energy to the object ball. (It is ironic that a clue to the answer has been in front of us all the time.)

In this cas, physics CONFIRMS anecdotal observation.

I suppose we should all go out and buy 6oz cue sticks!!! /ccboard/images/graemlins/grin.gif /ccboard/images/graemlins/grin.gif /ccboard/images/graemlins/grin.gif

Merry Christmas everybody!

The mathematical result surprised me, and it forces me to recant what I have said on previous posts that "it should not make a difference." It does, in fact, make a (small?) difference.

I apologize in advance for the use of metric units, but it makes the math so much easier. I will give the english equivalents wherever they help.

For the purpose of this analysis I will assume that

* The collision is perfectly elastic (energy and momentum will be conserved in the post collision motion of the stick and ball).

* The same shooter exerts a constant force (300 Newtons, about 67.4 lbs) over a 6 inch forward swing to impact. (The actual force and swing distance are not important here so long as they are the same for both cues.)

* The "light" stick weighs 18oz, the "heavy" stick weighs 21oz, and the cue ball weighs 6 oz.

* The stick and ball are free bodies in space. The the stick is accellerated to hit a ball that is initially at rest, and the axis and path of the stick is alligned with the center of the ball. (I know real balls and sticks are not free bodies in space, but this assumption simplifies everything without changing the outcome.)

The trick is to find the post collision speeds of the ball and stick so that energy and momentum are conserved. We have a system of two equations and two unknowns, so it can be solved with some hairy algebra. I used the Excel Solver routine because it was easier and I am LAZY.

A constant force of 300N applied over 6 inches will accelerate an 18oz (0.5103kg) stick to a speed of 13.38614 m/s (29.94mph). The pre-collision momentum of the stick is 6.830947 kg-m/s, and the kinetic energy is 45.72J. Conserving energy and momentum, the post collision speed of the stick is 6.69307m/s (14.97mph) and the post collision speed of the ball is 20.07m/s (44.92mph).

A constant force of 300N applied over 6 inches will accellerate a 21oz (0.5953kg) stick to a speed of 12.39368m/s (27.72mph). The pre-collision momentum of the stick is 7.377956 kg-m/s, and the kinetic energy is 45.72 J. Conserving energy and momentum, the post collision speed of the stick is 6.885376m/s (15.40mph) and the post collision speed of the ball is 19.27905m/s (43.13mph).

DRUM ROLL and TRUMPET FANFARE:::::::::::::

The cue ball has 4.1% more speed and 8.5% more energy when struck by a LIGHTER (18oz) stick than when it is struck by a HEAVIER (21oz) stick, all other things being equal. A 15% reduction in weight results in 4% more speed.

This is true - NOT because the lighter stick moves faster - but because the lighter stick transfers more energy to an object that is more similar in mass.

In fact, I demonstrated with the same spreadsheet that, as the mass of the two colliding objects gets more and more equal, more and more of the energy of the first object is transfered to the second object. When the mass difference is zero, the energy transfer is 100%.

You might find that hard to believe. Remember those annoying clickety-clackety novelty toys that had 5 steel ball bearings suspended on a wooden frame? Take three balls out of the way and try it with just two balls. The first ball will stop dead as the second ball takes 100% of the energy and swings upward.

We can also see this principle on the pool table. A dead-stratight stun shot transfers 100% of the cue ball energy to the object ball. (It is ironic that a clue to the answer has been in front of us all the time.)

In this cas, physics CONFIRMS anecdotal observation.

I suppose we should all go out and buy 6oz cue sticks!!! /ccboard/images/graemlins/grin.gif /ccboard/images/graemlins/grin.gif /ccboard/images/graemlins/grin.gif

Merry Christmas everybody!