二维弹性球碰撞物理 [英] 2D Elastic Ball Collision Physics

问题描述

我正在制作一个涉及弹性球物理的程序.我已经计算出与墙壁和静止物体碰撞的所有数学运算，但我无法弄清楚当两个移动的球碰撞时会发生什么.我有质量和速度(准确地说是 x 和 y 速度，但每个球的速度及其方向都可以)并且想要这些公式.记住 - 这是一个完全弹性的碰撞 - 所以没有旋转球等

解决方案

这篇

对于我们定义的每个球:

• 弥撒
• vi碰撞前的速度矢量
• v'i碰撞后的速度矢量
• Oi 中心点
• xi Oi 位置的向量

单位向量 n 垂直于球接触点的表面.

单位向量 t 在接触点处与球的表面相切.

<小时>

使用的物理定律

总动量守恒表示为:

总动能守恒表示为:

由于没有在切线方向施加力，碰撞后速度的切向分量不变:

<小时>

证明

速度的切向分量不变.所以我们可以用普通分量重写守恒定律，我们现在有一个一维问题:

动能守恒可以分解，然后用动量守恒简化:

我们将最后一个表达式与动量守恒结合起来，得到v'1的法向分量:

最后，我们找到了维基百科文章 v'1 的公式:

v'2的公式是对称的.

I am making a program that involves elastic ball physics. I have worked out all of the maths for collision against walls and stationary objects, but I cannot figure out what happens when two moving balls collide. I have mass and velocity (x and y velocity to be exact, but velocity of each ball and their direction will do) and would like the formulae for those. Remember - this is a perfectly elastic collision - so no spinning balls, etc.

解决方案

This wikipedia article provides a formula to compute velocities after collision between two particles :

There are many reasons to use this formula :

• you just need the velocity vectors of your balls before collision, their mass and their position,
• you don't need to define angles of deviation,
• the operations are simple (just dot product required),
• the vectors can be expressed in any coordinates system.

There is no proof in the wikipedia article so I provide it below.

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Definition of the problem

For each ball we define :

• mi the mass
• vi the vector of velocity before collision
• v'i the vector of velocity after collision
• Oi the point of center
• xi the vector of Oi position

The unit vector n is normal to the surfaces of balls at the point of contact.

The unit vector t is tangent to the surfaces of balls at the point of contact.

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Physics law to use

The conservation of the total momentum is expressed by :

The conservation of total kinetic energy is expressed by :

As there is no force applied in the tangential direction, the tangential components of velocities are unchanged after collision :

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Proof

The tangential components of velocities are unchanged. So we can rewrite the conservation laws with normal components and we have a 1D problem now :

The conservation of kinetic energy can be factorized then simplified with the conservation of momentum :

We combine this last expression with the conservation of momentum and we get the normal component of v'1 :

Finally, we find the formula of the wikipedia article for v'1 :

The formula of v'2 is symmetrical.

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