二维弹性球碰撞物理 [英] 2D Elastic Ball Collision Physics
问题描述
我正在编写一个涉及弹性球物理的程序.我已经计算出所有与墙和静止物体碰撞的数学公式，但是我无法弄清楚当两个运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，但是每个球的速度及其方向都可以)，并且想要这些公式.请记住-这是一次完全弹性的碰撞-因此没有旋转球等.
此维基百科文章提供计算两个粒子碰撞后的速度的公式:
使用此公式的原因很多:
- 您只需要碰撞前球的速度矢量，质量和位置，
- 您无需定义偏离角度，
- 操作简单(仅需点积)，
- 向量可以在任何坐标系中表示.
维基百科文章中没有任何证据，因此我在下面提供.
问题的定义
我们为每个球定义:
- 弥撒
- vi 碰撞前速度的向量
- v'i 碰撞后速度的向量
- 我是中心点
- xi Oi位置的向量
单位矢量 n 在接触点处垂直于球的表面.
单位矢量 t 在接触点处与球的表面相切.
要使用的物理法则
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上施加力，因此碰撞后速度的切线分量不变:
证明
速度的切向分量不变.因此，我们可以用正常的分量重写守恒定律，现在我们遇到了一维问题:
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 的正态分量:
最后，我们找到 v'1 的维基百科文章公式:
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.
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.
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.
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 :
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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