catalogue

First, the basic explanation of the problem.

2. 1. Research on barrier-free billiards.

2. Research on cutting the ball.

3. Research on the problem of eating warehouses

Three. Expand the application.

One. ? Basic explanation of the problem:

The problem we are discussing is billiards. Considering some characteristics of billiards and tables, we set the following basic assumptions.

1.？ Billiards have the same radius and the same mass.

2.？ The friction environment of billiard table top is the same.

3.？ For the time being, the horizontal rotation of billiards after hitting the ball is not considered, that is, billiards only roll in the direction of motion after hitting the ball.

So we have the following inference:

4. There is no "bouncing ball" event in the process of billiards impact.

As shown in the figure, under the premise of 1, it is easy to know that plane ABC is parallel to plane DEF. In the process of billiards collision, balls collide with balls and balls collide with the library on the plane ABC. Therefore, if not explained later, the plan describes the positional relationship on the plane ABC.

Two.

1. Research on barrier-free billiards.

The difference between rolling friction and sliding friction;

Sliding friction is the resistance caused by relative sliding between surfaces of objects. However, the rolling dynamic friction force is a component of the elastic force generated by the deformation of the supporting surface due to the extrusion of the object surface during rolling. (pictured)

According to Hypothesis 2 and Newton's second law, an object decelerates at a uniform speed.

Experiment:

1.？ Experimental purpose: to calculate the acceleration of billiards movement by using the formula of uniform deceleration movement.

2.？ Equipment: two stopwatches, tape measure, billiard cue, calculator.

3.？ Specific operation: (1) Use two billiard sticks placed in parallel as "slideways", and place an unfolded tape measure at the ends of the billiard sticks along a straight extension line.

(2) Two students, one standing at the end of the tape measure and the other standing at the midpoint of the tape measure. Each of them has a stopwatch.

(3) Roll the billiards down from any height (high enough for the ball to cross one end of the tape measure), and both of them will start timing at the moment when the billiards roll down the "title" and stop timing when the ball rolls over one end and the other.

(4) Record the half-length l of the tape measure, and the two timing results t 1, t2.

According to the formula of uniform deceleration motion, it is not difficult to deduce: a=?

Number of tests? l？ (cm)？ t 1？ t2？ a

1? 70? 1. 16s？ 2.25s？ 0.078

2? 70? 1.68s？ 3.59s？ 0.02

3? 60? 1. 16s？ 2.34s？ 0.00749

4? 60? 1.42s？ 3.26s？ 0.059

Data:

Considering the serious deviation of two or three data in the above experiment, the data of 1 and 4 are quite consistent. Our analysis may be that the timing method of two people has increased the error.

So we improved the experimental method. Measure the stopping time and stopping distance of the ball, and calculate the acceleration with S = 0.5at2 ..

Number of tests? S stop? Don't stop. a

1? 69? 3.75 seconds? 0.098

2? 1.8? 5.8 1? 0. 1 1

3? 0.83? 4s？ 0. 103

The results of this experiment fluctuate very little, so we combine the results of this experiment to calculate the analytical formula of ball motion.

=(0.098+0. 1 1+0. 103)? =0. 103?

So there is: v = v0-0.1t.

S=v0t-？

x=5v02

2. Research on cutting the ball

Inference is made on the premise of basic assumptions 3 and 4. We introduce "momentum theorem" to study.

( 1)? Momentum theorem: In a mechanical system, if the system is not affected by external force, its total momentum remains unchanged. Momentum is the product of mass and velocity.

(2)? Momentum Theorem Inference: When an object is stationary and two objects collide concentrically with the same mass, the velocity of the "hit object" after collision is equal to the original velocity of the "hit object". (Physics Elective 5-5)

(3)? Research on center collision: We know that billiards move at a uniform speed under the condition of five obstacles, but the speed at the moment of collision can represent the original speed of the "hit object", and the initial speed obtained by the hit billiards can represent the speed obtained by the "hit object".

Namely: v0/=vt

(4) Research on the non-concentric collision:

See the picture on the next page for the positional relationship between the hole, cue ball and colored ball.

Since momentum is a vector, it can also be decomposed.

As shown in the figure,

Observe the position relationship diagram of the hole, cue ball and colored ball.

In the figure, if the distance AD between two balls and the distances CD and AC between balls and holes are known, the triangle ACD can be solved. If we know the radius r of the ball again? The triangle ABD can be further solved.

Solution: cos?

Because?

AB2=2AD2+4R2+CD2-AC2

∴sinB=BD？ =2R？

Considering that simplification is of little significance, the above is the final result.

The speed of the ball after being hit is: V/=v? cos∠ADB？

Here we have done an experiment to verify that the friction on the surface of billiards has no effect on the "tangent direction" of billiards.

Test: 1. Tools: white paper, cue, slide, billiards, ballpoint pen.

2。 Operation: Put the white paper in the center of the desktop, draw a "marking point" on the paper, and put the billiards on the tangent point. Let the white ball roll down from a fixed height and play billiards from the side. (as shown in the figure below)

Record the position of the colored ball when it is knocked off the edge of the white paper, and then add the "marking point" back. Change the height of billiards on the slide, do several groups of experiments, record the position of each color ball when it is hit off the edge of white paper, and make a comparison. ?

Phenomenon: the position of the colored ball remains unchanged when it is knocked out of the edge of the white paper.

Conclusion: The surface friction of billiards has no effect on the tangent direction of billiards.

For this fact, we need a more accurate theoretical explanation:

In fact, it is not difficult to imagine that for a rolling sphere, the speed of each point on its "impact surface" is perpendicular to the horizontal plane (with the center of the sphere as the reference). As shown below. The arrow represents speed.

All points on the impact plane have the same horizontal velocity and different vertical velocities.

So when two balls collide, the white ball decomposes its horizontal velocity into "normal velocity" and "tangential velocity"; Pass them to the colored balls respectively, and the friction will only transfer the "tangential speed". ? The normal velocity is transmitted to the ball in the form of impact. Suppose the angle between the normal velocity and the horizontal velocity is? Chiichi, when? When it increases, the normal velocity decreases and the tangential velocity increases. The extrusion force at the contact point is reduced, so that the friction force is reduced. On the contrary, when? When it decreases, the normal velocity increases and the tangential velocity decreases. The moving tendency of the contact point decreases, thus reducing the friction effect.

It can be seen that friction will have a certain impact, but its impact has always remained at a low level, which is difficult to detect macroscopically.

Then we have to consider the influence of vertical velocity. In fact, the friction generated by this speed is perpendicular to the desktop, which will have a certain "braking effect" on the colored balls. It has no effect on the direction of the ball.

4.？ Research on the problem of eating warehouse;

Experiment: Objective: To verify that the friction between billiards and the library surface will affect the reflection direction of billiards.

Operation: Spread a piece of white paper on one side of the table, set up a slide on the opposite side, let the reflection path of the ball bounce on the paper, record the positions of the entry point, reflection point and exit point, mark it on the paper, and use a ruler? The connecting line represents the trajectory of the ball, and the normal line is made by dotted line. Repeat the experiment.

Phenomenon: The incident angle is greater than the reflection angle.

Conclusion: The friction between billiards and library surface will affect the reflection direction of billiards.

Next, we carry out theoretical analysis and derive specific formulas.

First of all, let's analyze the reasons for this phenomenon:

In the process of billiards hitting the library, its momentum can be decomposed as shown in the figure. In the process of head pass, the acceleration of the ball is extremely high, and the friction between the ball and the library can not be ignored. This friction forces the sphere to rotate, so we can assume that the linear velocity of the great circle is equal to Vx. After the rotation occurs, part of the original kinetic energy is converted into angular kinetic energy and then into internal energy (due to the friction of the desktop). If the kinetic energy is less, the natural reflection angle will also be affected.

Here we need to introduce the calculation method of angular kinetic energy to help the operation.

The angular kinetic energy of an object rotating around a fixed axis is E=? Iz？

Where Iz is the moment of inertia, and the values of different objects are different.

By looking up the table, we get: What is the moment of inertia of the sphere? , substituted into the above formula,

E angle =? =? . V is the linear velocity of the great circle.

Let the incident angle be? What is the reflection angle? .

Start with e =? mv2

⊿E=？ E angle =?

E-end? =? Start with -⊿E=? =?

So, v/=

Because the texture of the library is hard, it can be considered that momentum in the y direction is conserved.

So, cos? =?

Because? Is greater than 1. So what? Greater than? , consistent with the conclusion.

Expand application: First of all, this paper can help billiards players make analysis during training. It can accurately calculate the goal rate of athletes in a certain situation.

Secondly, it can be applied to the sports analysis of billiards, such as sand ball, curling ball, hockey and so on.

Thirdly, it can be used in microphysical experiments: for example, adjusting the trajectory of particles to reach a predetermined position.

Fourth, we can analyze the process and results of planetary impact.