Source: http://www.flickr.com/photos/ventsislav/2222807833Physics of Billiards – sphere CollisionThe physics behind billiards (or the physics behind pool), in huge part, entails collisions between billiard balls. When two billiard balls collide the collision is almost elastic. An elastic collision is one in which the kinetic power of the mechanism is conserved before and also after impact. Therefore, because that simplicity one have the right to assume the for collisions entailing billiard balls, the collision is perfect elastic.For collisions in between balls, inert is constantly conserved (just prefer in any type of other collision). For a simplified situation assuming no friction (discussed below), us can incorporate this fact with the elastic-collision presumption to find the trajectory of 2 colliding billiard balls after ~ impact. The figure listed below shows a collision between two billiard balls. Because that the basic case, the collision is not head on, which is what the number shows.It is assumed that balls A and also B have actually the same mass and that ball B is originally at remainder (zero velocity). The early velocity of ball A is V1A. After ~ impact, ball A move at velocity V2A in the direction shown, and ball B moves at velocity V2B in the direction shown.

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The heat L1 is drawn at a tangent to both balls in ~ the point of contact. Because of geometry, L1 is perpendicular to the heat passing with the facility of the two balls and the contact suggest CP. Due to geometry, L1 likewise makes an angle θ through the vertical, and the line passing with the center of the balls renders an angle θ v the horizontal.After influence at CP, round B moves in the direction of the heat joining the facility of the two balls, together shown. This is since the pressure (impulse) delivered by round A to sphere B acts typical to the surface ar of round B, assuming over there is no friction in between the balls (a good assumption since billiard balls are smooth). Thus, ball B moves in the direction the this impulse.Notice that, after impact, ball A move in a direction perpendicular come the direction of ball B. This interesting result can be proven as follows. Analysis Of round Collision
For the 2 colliding balls, the basic vector equation because that conservation of direct momentum is:
Since the masses mA and also mB are assumed equal, this equation simplifies to:
For an elastic collision kinetic energy is conserved, and also the equation is:
Since the masses mA and mB are equal, this equation simplifies to:
By the Pythagorean theorem, this critical equation tells us that the vectors V1A, V2A, V2B kind a right-angled triangle. Therefore, the vector equation for conservation of momentum can be attracted as displayed below.
Thus, after affect ball A moves in a direction perpendicular to the direction of ball B. This is a really slick result.There space two extr special instances to consider, including ball collision.For the situation where the target sphere B should be hit in ~ an edge θ very close come zero (such as to sink that in the next pocket), sphere A requirements to be relocating at a high speed V1A (meaning you would need to hit ball A quite difficult with the cue). This is since only a very small portion of the momentum of ball A (and thus velocity) is transferred to sphere B, due to the obliqueness the the impact. For the instance where the influence is head ~ above (θ = 90°) the over solution does no apply. In this case V2A = 0 and V2B = V1A. This essentially method that the velocity of sphere A is fully transferred to sphere B.For a an ext detailed and complete analysis, in which the trajectory of sphere A is calculated (after impact), under the affect of friction in between the ball and also billiard table, watch the problem, Cue ball trajectory through table friction.The Sweet Spot
The physics the billiards is similar to the Physics the Hitting A Baseball, in that there is also a sweet point out on a billiard ball where you can strike v the cue stick so the no friction pressure develops in between the ball and also the billiard table. Understanding the location of this sweet spot can offer you an idea of wherein to fight the round so the it establishes backspin or front spin, which deserve to be useful when do a shot.Consider the figure listed below showing the place of the cue at height h.
We great to uncover the elevation h so that no (horizontal) frictional force develops at suggest P as soon as the round is win by the cue. Evaluation Of The Sweet Spot
In this analysis, we can represent the ball + cue mechanism with a free-body chart as presented below.
Where:F is the pressure the cue exerts on the ball as soon as it strikesr is the radius the the ballG is the center of massive of the ballg is the acceleration due to gravity, i beg your pardon is 9.8 m/s2P is the suggest of call of the round with the billiard tableFPx is the x-component the the force exerted on the ball by the billiard table, at suggest P. This is a frictional force.FPy is the y-component the the pressure exerted on the round by the billiard table, at point P.By Newton's second Law, the general force equation in the x-direction is:
Where:m is the mass of the ballaGx is the acceleration that the center of massive in the x-directionThis equation becomes
Since FPx = 0 we get
By Newton's 2nd Law, the general pressure equation in the y-direction is:
where aGy is the acceleration of the center of massive in the y-direction.Since the billiard ball just moves in the x-direction aGy = 0, for this reason the above equation becomes
Therefore
We must now write the general moment equation for rotation of a strict body about its facility of massive G.
Where:ΣMG is the sum of the moments about the facility of fixed GIG is the moment of inertia that the ball around its center of mass, around an axis pointing out of the pageα is the angular acceleration the the ballSince no frictional pressure develops in between the ball and table, over there is no loved one slipping at suggest P. This way that we have a situation of pure rolling. Thus, we can write the following:
In the above equation the negative sign is there to enhance the sign convention used in this problem.The minute equation becomes
Combine equations (1) and (2) and also we get
For a solid sphereTherefore
This is the height to struggle the ball so that no friction establishes at suggest P. No matter how hard you hit the sphere at this location, no friction (reaction) pressure will construct at suggest P. Therefore, pure roll of the ball will always result after influence (no loved one slipping).In the situations where the cue strikes above or below this height h, friction is essential to stop the sphere from slipping on the surface of the billiard table. And if the ball is hit hard enough (above or listed below height h) family member slipping will occur, because of insufficient friction in between ball and also table.In the cases where slipping occurs we have actually the following inequality:
This means that over there is family member motion between the ball and billiard table at point P instantly after impact. In other words (immediately after ~ impact), the tangential velocity of the sphere at point P is no equal in magnitude and also opposite in direction to the velocity the the center of massive of the round G.In the situation of pure rolling, the tangential velocity of the ball at allude P is same in magnitude and opposite in direction come the velocity that the facility of massive of the sphere G. Therefore, the velocities cancel out and there is no relative slipping at point P. Family member Slipping Analysis
Relative slipping in between ball and billiard table is one interesting point of analysis. It's informative to understand just how the round moves relying on where the is hit relative to h. Consider the number below.
When the sphere is hit sufficiently hard with a leftward force in an ar A1 the sphere is provided a leftward velocity, and also is given a backspin in the CW direction. Loved one slipping occurs at suggest P, and the resulting frictional pressure at this location is pointing right. The leftward velocity of the sphere decreases and it speeds up to the right due to the direction that the friction force. The rate of backspin decreases as result of the direction that the frictional force. This wake up until loved one slipping at point P stops and pure roll occurs.When the round is struggle sufficiently tough with a leftward pressure in region A2 the round is offered a leftward velocity, and also is provided a front spin in the CCW direction. Family member slipping occurs at suggest P, and also the resulting frictional pressure at this ar is pointing right. The leftward velocity of the round decreases and also it increases to the right as result of the direction of the frictional force. The price of front spin increases as result of the direction that the friction force. This occurs until relative slipping at point P stops and pure rolling occurs.When the sphere is fight sufficiently tough with a leftward force in region A3 the ball is offered a leftward velocity, and also is given a forward spin in the CCW direction. Loved one slipping occurs at allude P, and also the resulting frictional pressure at this location is pointing left. The leftward velocity the the sphere increases and it increases to the left due to the direction of the frictional force. The rate of forward spin decreases as result of the direction that the friction force. This occurs until loved one slipping at allude P stops and pure rojo occurs.Thus, the nature the the slipping will readjust depending on i beg your pardon of the regions, (A1, A2, A3), the cue rod strikes the ball.Note that for the three instances above, the frictional force that establishes due to family member slipping is known as kinetic friction. Kinetic friction occurs once there is "rubbing" in between two surfaces. This form of friction constantly opposes the direction that motion. So because that example, if a crate is sliding on the floor in the left direction, the kinetic friction between the crate and also floor acts to the right. In the instances where we have this form of friction you have to account because that the direction of loved one slipping and then entrust the direction the friction to it is in in the opposite direction. On the other hand, as soon as there is no loved one slipping in between two surface the frictional force between them is known as revolution friction. For this reason in the general situations where the billiard round experiences no family member slipping at point P, we have actually static friction maintaining pure rolling. Accountancy for static friction different from kinetic friction in that you don't require to recognize the direction in which it acts. The direction is addressed for in the dynamics equations. So in this feeling it is simpler to account for revolution friction 보다 kinetic friction.A Closer look at At family member Slipping (optional)
This is a continuation of the previous section, through a an ext in-depth look at at family member slipping. This ar is optional, for this reason you might skip the if friend like.The figure listed below shows a free-body diagram of a billiard sphere experiencing a general instance of family member slipping.
Where:VG is the velocity that the center of fixed of the ball. (In helpful terms, when one refers to the velocity that the sphere he is introduce to the velocity of the facility of mass of the ball).w is the angular velocity that the ballLet wi stand for the initial angular velocity of the ball instantly after impact.Let VGi represent the early velocity that the ball automatically after impact.With no lose of generality we can assume VGi is to the left (negative, according to the sign convention) and also wi is one of two people CW (negative) or CCW (positive).If the cue strikes the ball in an ar A3 and relative slipping occurs, thenwir > -VGi.If the cue strikes the sphere anywhere listed below height h and relative slipping occurs, climate wir VGi.Set
This is equal to +1 or -1. This element accounts because that the direction of loved one slipping, i beg your pardon is crucial to know due to the fact that we are managing kinetic friction. Such determinants are mathematically very convenient when audit for the direction that kinetic friction. (Note the |x| means the absolute worth of x).From before
This is the normal force acting ~ above the round at point P.The general force equation in the x-direction is:
The kinetic friction acting at ns is offered by:
where μk is the coefficient of kinetic friction between the ball and table.Now,
The basic moment equation for rotation the a strictly body about its facility of fixed G is:
This becomes
From equations (3) and also (4) us get:
The direct velocity the the round is:
where t is timeThe angular velocity the the round is:
We great to find the time the takes because that the ball to protect against slipping, and begin pure rolling. Thus, using equations (5) and (6) we can formulate the following equality, i m sorry holds true as soon as there is pure rolling:
From this we have the right to solve because that time t.

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For a solid sphereTherefore
Let's now find the distance traveled by the ball before pure rolling begins, using the moment t indigenous above.The street d is:
A final factor to consider is detect the velocity of the ball before pure rojo begins. To execute this us calculate the velocity VG using the time t native above.
Note the the over three equations are only valid as long as there is loved one slipping at suggest P.Sample calculation For loved one SlippingThis is a sample calculation utilizing the results of the ahead section.VGi = -1 m/s (initial leftward velocity)wi = -20 rad/s (backspin, CW rotation)r = 0.028 mμk = 0.3g = 9.8 m/s2δ = 1aGx = 2.94 m/s2 - calculated indigenous equation (3)Therefore the moment it take away the sphere to stop slipping ist = 0.15 sThe distance traveled by the ball during slipping isd = -0.12 m (the ball has traveled come the left)The velocity of the round just before pure rolling starts isVG = -0.56 m/s (leftward velocity)Closing Remarks
As you can see, the physics of billiards can get pretty affiliated when you begin considering all the things that can take place in a usual game that pool. You have the right to bet that expert players are an extremely proficient in the practical consumption of the physics gift here, and also other elements of the game not discussed here.Return come The Physics of Sports pageReturn come Real people Physics Problems residence page