# I A ball struck by a cue in billiards with English goes straight at first...

#### poolplayer

Hi, let me ask probably dumb questions to physics experts... Why does the cue ball goes almost straight when its right (or left) side is struck by a cue (right English)? This is quite different from when a ball hits another ball, in which case the ball goes almost perpendicular from the contact surface. Here, I want to focus on the ball direction immediately after the impact and ignore the curve after the ball starts to roll.

I heard that it is because the ball and cue can be thought as unity so the force only propagates to the cue direction. But if so, would it be possible that the cue makes the ball spin? My guess is that the friction between the ball and cue tip makes the ball not only spin but also go to the right direction and counteracts the force perpendicular from the contact surface (left), resulting that the ball goes to the cue direction? I heard that the friction coefficient is pretty high ~0.6. Any comments will be appreciated!

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#### fresh_42

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The ball first slips over the cloth, with or without rotation and independent of where it was hit. Once it hits another ball friction takes over and the linear movement becomes a rotation, such that the ball rolls from there on. During the collision, the linear movement comes to a hold for a moment which forces the existing angular momentum into a rotation.

#### poolplayer

Thank you for the comment! I didn't know that a ball always slips after I hit the ball. Would it be also the case when I strike the ball right above the center very softly? And I still wonder why the ball doesn't go to the left when I strike its right side with a cue stick...

#### poolplayer

It surely goes to left a little bit, but it goes relatively straight (the direction of the cue stick).

#### fresh_42

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And I still wonder why the ball doesn't go to the left when I strike its right side with a cue stick...
If you play the ball properly, it receives a single force in the direction of the cue. For a change in direction you would need another force, either from a different direction or in addition to the cue's force. It happens if your cue slips off or along the ball which usually is a foul.

#### poolplayer

But, if a cue ball collides with the right side of an object ball, the object ball goes to the left, right? What is the difference between the ball-ball collision and ball-cue collision? I am sorry, but I am really weak at physics... I guess this is a very basic question.

#### poolplayer

To make it clear, I posted a drawing. Why does a ball go to different direction between ball-ball and cue-ball collision?

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#### Cutter Ketch

The answer is indeed friction. The acceleration is not parallel to the normal of the contact point, so there must be a tangential component to the force. That requires friction or adhesion. The cue tip is soft compared to the balls and the contact lasts much longer than a ball to ball collision. The cue tip deforms, effectively sticks to the ball, and forces it along in the same direction the cue tip is traveling. It is something like throwing the ball rather than a hard elastic collision. During the contact with the cue the cue ball does indeed slide across the felt. This is easy to see if you look for it.

The need for high friction, or even adhesion is the purpose of chalk. A miscue is the result of the cue slipping along the surface of the ball. This happens when the friction is low due to lack of chalk, the contact is poor due to a poorly shaped, worn, or hard tip, or the friction required exceeds the friction available because the ball was struck too far off center and the tangential component is too large. Even with sticking the ball will deflect some. I've seen claims that say the stickier the chalk the less the cue ball deflects off axis. However that doesn't make sense to me. Either the cue tip slips or it doesn't. Once you've avoided slipping more stickiness is not any better.

The critical difference with the collision of two balls is that they are very hard and elastic and the contact lasts a very short time. There is a little bit of friction and some nonneglible throw in ball to ball collisions. However they are much closer to an ideal elastic collision and the direction is usually very close to the normal. On the other hand, accounting for the throw particularly at thin angles is part of what separates a good player from the rest of us.

#### fresh_42

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But, if a cue ball collides with the right side of an object ball, the object ball goes to the left, right? What is the difference between the ball-ball collision and ball-cue collision? I am sorry, but I am really weak at physics... I guess this is a very basic question.

On the right side the red line shows the part of the force that applies to the object ball. The rest of the momentum is carried away by the white ball whereas the cue doesn't take any momentum to a different direction.

#### Cutter Ketch

View attachment 110621

On the right side the red line shows the part of the force that applies to the object ball. The rest of the momentum is carried away by the white ball whereas the cue doesn't take any momentum to a different direction.
The cue has a constraint and experiences external forces. Momentum conservation is not relevant. If you took the cue ball in hand and rammed it straight ahead through the object ball without deflecting I believe you would find that the object ball still goes off at an angle. (Somebody with a pool table please try). I don't think this is why the cue is different.

#### poolplayer

Thank you for your answers! I am still trying to understand the insights into momentum conservation (if so, is cue ball direction different when hit by rolling ball and ball-in-hand?). But friction explanation makes a lot of sense to me.

You know cue makers sell "low-deflection" cues with low density (mass) at the cue tip. Striking with a low-deflection cue with right English, a cue ball goes even more straight with less deflection to left direction. Is this probably due to increase in contact time between the cue and ball (because the cue is softer) and more friction*time to the right direction (and maybe less coefficient of restitution reduces the cue ball deflection to left)?

My thought is that if it is friction that makes the cue ball go straight, it would be theoretically possible to make the cue ball to the same direction as English if friction force can be increased (one could strike the right side of a ball and the ball goes to the right).

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#### poolplayer

If you took the cue ball in hand and rammed it straight ahead through the object ball without deflecting I believe you would find that the object ball still goes off at an angle.
I am pretty sure that the ball goes off though I haven't tried it.

#### Cutter Ketch

Thank you for your answers! I am still trying to understand the insights into momentum conservation (if so, is cue ball direction different when hit by rolling ball and ball-in-hand?). But friction explanation makes a lot of sense to me.

You know cue makers sell "low-deflection" cues with low density (mass) at the cue tip. Striking with a low-deflection cue with right English, a cue ball goes even more straight with less deflection to left direction. Is this probably due to increase in contact time between the cue and ball (because the cue is softer) and more friction*time to the right direction (and maybe less coefficient of restitution reduces the cue ball deflection to left)?

My thought is that if it is friction that makes the cue ball go straight, it would be theoretically possible to make the cue ball to the same direction as English if friction force can be increased (one could strike the right side of a ball and the ball goes to the right).
If the ball completely completely stuck to the cue (like fused with it or something) it would travel straight with the cue. I think straight with the cue is the limit. I don't think you can make the ball turn towards the English during the collision. Of course if you can get the ball spinning correctly you can make it curve that way after the collision.

#### sophiecentaur

Science Advisor
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On the right side the red line shows the part of the force that applies to the object ball. The rest of the momentum is carried away by the white ball whereas the cue doesn't take any momentum to a different direction.
That comment worries me. Momentum (linear and angular) still has to be conserved. The cue / arm / rest of the world have a momentum change due to the collision. The resulting velocity change of the cue is 'near' zero because the effective mass is so high.
This would happen for a collision in space, with no contact with the ground. Whatever happens, due to friction etc., is a separate issue - total Momentum will still be conserved during the spinning interaction. The initial motion (Impulse) will be to the left but any friction effect of the ball spinning will be in addition to this and happens after the contact with the cue.
PS What is "turning to the English"?

#### poolplayer

Cutter Ketch said:
I think straight with the cue is the limit.
I think you are right... Is this because the friction force is product of friction coefficient (<1.0) and the force perpendicular to the impact surface?

I was wondering whether the softness of the cue tip could reduce the initial velocity of the ball to left by reducing restitution coefficient or spring constant so that the friction force dominates and ball goes to right... I also noted that the cue could be bent to right during the impact (right English). I am not sure whether this bend significantly affects the ball direction though (I guess the bend increases both friction force to right and bouncing force to left)

sophiecentaur said:
What is "turning to the English"?
It refers to what I said: when a ball was hit on the right side (right English), the ball goes to the right, which may be impossible.

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#### sophiecentaur

Science Advisor
Gold Member
It refers to what I said: when a ball was hit on the right side (right English), the ball goes to the right, which may be impossible
Right hand side then?
If it is struck on the right hand side then it (the centre of mass of the ball) cannot move to the right initially. It has to move towards the left. That is what momentum conservation tells us and there will be a force on the cue tip, to the right. The tip may bend or be rigid and the details of what happen to the ball will depend on the stiffness of the cue etc. and also the friction of the tip on the ball and the ball on the table. There is no simple answer but friction will make the ball rotate anticlockwise (looking down on it) and, depending on whether the contact is above or below the mid point, the ball will also have top spin or bottom spin. So the path of the ball can follow a right hand or left hand curve, depending.
The whole business of striking the ball with the cue will have many variables involved and what the player 'thinks' is happening may well not be what is actually happening. That doesn't matter to the player, who learns, from experience and from 'discussions' what to do. The Physics may be very loosely connected with the player's subjective feeling about what's gong on. This doesn't matter; a good player is a good player and doesn't have to be a good Physicist.

#### Cutter Ketch

I think you are right... Is this because the friction force is product of friction coefficient (<1.0) and the force perpendicular to the impact surface?

I was wondering whether the softness of the cue tip could reduce the initial velocity of the ball to left by reducing restitution coefficient or spring constant so that the friction force dominates and ball goes to right... I also noted that the cue could be bent to right during the impact (right English). I am not sure whether this bend significantly affects the ball direction though (I guess the bend increases both friction force to right and bouncing force to left)

It refers to what I said: when a ball was hit on the right side (right English), the ball goes to the right, which may be impossible.
When I said the ball couldn't go the other way I hadn't thought about the idea that the cue may deflect. However, under the assumption the the player is trying to move the stick in a straight line and any deflection is a reaction to hitting the ball, I would think that would move the ball off more in the expected way by Newton's law of action and reaction.

Now if the player jerks the tip the outside direction during the collision that is a different story.

#### sophiecentaur

Science Advisor
Gold Member
When I said the ball couldn't go the other way I hadn't thought about the idea that the cue may deflect. However, under the assumption the the player is trying to move the stick in a straight line and any deflection is a reaction to hitting the ball, I would think that would move the ball off more in the expected way by Newton's law of action and reaction.

Now if the player jerks the tip the outside direction during the collision that is a different story.
You are neglecting the question of top spin and bottom spin. All good players are well aware of the effects of these on the path of the ball.

#### poolplayer

Cutter Ketch said:
under the assumption the the player is trying to move the stick in a straight line and any deflection is a reaction to hitting the ball, I would think that would move the ball off more in the expected way by Newton's law of action and reaction.
So, the bend would make the ball goes even more left (more cue ball deflection). I don't know its math, but I believe you. I will do a little research and find how I can calculate them. Thank you for your suggestions!

sophiecentaur said:
a good player is a good player and doesn't have to be a good Physicist.
Right, but I wanted to know basic Physics as well! So, do you think that the friction only affects the curve and does not affect initial direction? Curve is of course an important factor in the game. But I will think about curves in other time, and here I want to focus on the initial direction/velocity of the cue ball.

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#### poolplayer

Cutter Ketch said:
Now if the player jerks the tip the outside direction during the collision that is a different story.
This is a good point. Some people do this hoping to reduce cue ball deflection.

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#### Cutter Ketch

You are neglecting the question of top spin and bottom spin. All good players are well aware of the effects of these on the path of the ball.
If you read the last sentence of post 13 you will see I am not neglecting spin. However I do believe spin acts to change the trajectory primarily after contact with the cue is ended.

#### sophiecentaur

Science Advisor
Gold Member
If you read the last sentence of post 13 you will see I am not neglecting spin. However I do believe spin acts to change the trajectory primarily after contact with the cue is ended.
PF General Physics forum has a regular stream of questions where a 'sporting' phenomenon is analysed (boxing, tennis, motor sport etc). It always gets out of hand very quickly because the processes don't ever seem to have enough measurement data available. The problem which always arises is to decide how 'ideal' a situation is to be analysed.
So a perfectly elastic frictionless ball on a frictionless table would not spin at all and the only force would be normal to the ball. The direction of motion would be along a line which is a radial to the ball from the contact point. That's the starting point and the simplest case. Now how far do you want to go in introducing practical factors? Which ones would you include and which would you ignore? You'd have to include a lot of those factors and then you could perhaps show that some of them are not significant.
Once the contact time is not assumed to be instantaneous, it gets harder. If the ball or cue tip deforms then the contact time would be finite and the amount of rotation would depend on the length of time of contact and the lateral force over that time (and the MI of the ball).

#### Cutter Ketch

You are neglecting the question of top spin and bottom spin. All good players are well aware of the effects of these on the path of the ball.
In quoting the above I remind you that you are the one who wanted to talk about spin.

#### poolplayer

The direction of motion would be along a line which is a radial to the ball from the contact point.
In case you doubt that the cue ball goes straight with right English (struck right hand side), I put a little movie (20-times slower than actual time). You can see the cue ball direction (yellow line) is far from normal to the contact surface (red line). So some significant force should be taken into account. I now think that it is friction, although I am not sure how strong it is with the relatively short contact time (a few milliseconds at most, typically 1 ms). There may be other significant components. You can see that the cue waggles a lot after impact, indicating significant cue deflection.

Billiards are probably the simplest sports in terms of physics and I don't see any obstacles to measure everything in Billiards.

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#### Cutter Ketch

In case you doubt that the cue ball goes straight with right English (struck right hand side), I put a little movie (20-times slower than actual time). You can see the cue ball direction (yellow line) is far from normal to the contact surface (red line). So some significant force should be taken into account. I now think that it is friction, although I am not sure how strong it is with the relatively short contact time (a few milliseconds at most, typically 1 ms). There may be other significant components. You can see that the cue waggles a lot after impact, indicating significant cue deflection.

Billiards are probably the simplest sports in terms of physics and I don't see any obstacles to measure everything in Billiards.
Terrific stuff. Thanks for posting that. I wish my iPhone could scroll through frame by frame better. Well, tomorrow when I'm at the computer ...

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