The physics of billiards, the physics of billiards

This problem goes a step further than most of the problems in this book, and it involves the problem of the conservation of both momentum and kinetic energy, while also using the vector method. If an inexperienced man plays and he successfully hits the "8" ball into a pocket in one corner, is there a great danger that the "Q" ball will be put into the pocket on the other corner at the same time? (If the "Q" ball falls into the pocket, there is a penalty point.)

a) is likely to result in a penalty ball in the position shown;

b) There is very little danger of causing a penalty ball in the position shown.

A. Every billiards player knows that when the "Q" and "8" balls collide, the two balls bounce off at an angle of about 90 degrees, that is, they are separated roughly at right angles to each other. In the picture, the two corner bags are separated by an angle of 90 degrees relative to the position of the "8" ball, so the danger of the "Q" ball falling into the bag is very great.

Why do balls separate at right angles? The two balls have (or should have) equal mass, so their momentum should be proportional to their velocity. Therefore, the sum of the velocities of the "8" and "q" balls after the collision should be equal to the velocity of the "q" ball before the collision.

But as you can see from the diagram, there are many possible combinations where the sum of the two velocities is equal to the starting velocity of the "q" ball. Which one should you choose?

Not only momentum but also energy should be considered in this question. Because the sum of the kinetic energy of the two balls after the collision is close to equal to the kinetic energy of the 'q' ball before the collision. The kinetic energy of a ball is proportional to the square of its velocity.

Since the two balls have the same mass, the square of the velocity of the "q" ball after the collision plus the square of the velocity of the "8" ball should be equal to the square of the velocity of the "q" ball before the collision. According to the vector addition rule, it can be seen that the velocity vectors of the "8" and "q" balls after collision constitute the two sides of the parallelogram. According to the law of conservation of momentum, the diagonal of the parallelogram is equal to the original velocity vector of the 'q' sphere.

From the law of conservation of kinetic energy, it can be seen that the sum of the squares of the two sides of the parallelogram must be equal to the square of the diagonal. This means that the angles formed by the adjacent sides of this particular parallelogram must be right angles! Remember Pythagoras' law of triangles?

From this, it was concluded that the two balls bounced apart at an angle of 90 degrees. Why do we leave room to separate at exactly 90-degree angles, but roughly at 90-degree angles? Because the collision is not a fully elastic collision, part of the original kinetic energy becomes heat.

Also, there is friction between the balls and between the balls and the table, so the momentum and energy of the balls are not strictly equal before and after the collision. Some of the energy may also become the rotational energy of the ball after the collision. Experienced billiards players have taken advantage of these effects to find a way to pocket the "8" ball with a "Q" ball without being deducted points.

The action of the force is reciprocal, and in the absence of an external force, the colliding object is subjected to a force of the same magnitude in the opposite direction.

Collisions in physics are divided into two categories: fully elastic collisions and non-fully elastic collisions. A fully elastic collision is an idealized collision – in which there is no loss of energy. Collisions between objects made of hard materials are generally regarded as fully elastic collisions, such as between steel balls, glass balls, steel balls and hard ground, etc.

There is an energy loss in non-fully elastic collisions, which is also a common type of collision. In the event of a non-fully elastic collision, if two objects in the collision stick together after the collision, this kind of collision is called a completely inelastic collision, and its energy loss is the largest.

Whether it is a fully elastic collision or a non-fully elastic collision, they follow the law of conservation of momentum. The law of conservation of momentum is more applicable than Newton's law of motion, and it applies not only to the interaction between stars in the universe, but also to the interaction between elementary particles in the microscopic world. When two objects collide, there are two forms of positive collision and oblique collision.

For billiards, in the process of hitting, according to the position of the main ball and the target ball, the hitting method of forward and oblique collision is basically adopted. In the way of hitting the oblique ball, it is also necessary to choose the angle at which the cue ball collides with the target ball according to the need, which is a skill that must be mastered in billiards.

According to the momentum theorem, the velocity of the cue ball after the cue is struck is related to the amount of impulse given to the ball by the cue. The impulse of the club is determined by the amount of force applied to the ball by the club during the blow and the duration of the action. In the momentum theorem:

When the object is subjected to the impulse i, it causes the momentum p of the object to change, and its expression is i = δp, i.e., fδt = δp. In order to make the velocity of the target ball change large (but the momentum change is certain), two basic methods can be adopted: one is to apply a large striking force through the cue, but the hitting time is shorter; The second is to apply a smaller striking force through the club, but the striking time is longer. The striking force of the club can be controlled, and the hitting time is achieved by manipulating the club.

If the billiards player hits violently and stops after hitting, then the hitting time is very short; If the billiards player applies the force evenly and allows the cue stick to continue to extend in the direction of the force, then the batting time will be longer.

It's best to calculate this thing yourself.,This will be more impressive.。。 I won't.. The technical aspects are:

Hitting the top half of the cue ball will have a tendency to follow the target ball, and hitting the bottom half will prevent the cue ball from following the target ball. Spinning the ball is to hit the upper left or upper right part of the cue ball, which will produce the effect of a curved ball like a free kick in football, and the forward path of the cue ball is arc-shaped, which can bypass other balls blocked in a straight line in front and hit the target ball, which is difficult to practice.

Check out this fancy shot