评估与墙壁碰撞后的刚体线速度和旋转速度

Question

我有一个相对简单的案例，我已经花了很长时间，我正在尝试手动评估（不使用任何库）二维正方形的旋转速度和线性速度和撞墙后的球。

鉴于：

r - is either radius or side length of the square
X1 - is a vector of initial position for a given rigid body
V1 - is a vector representing initial velocity of a given rigid body
R1 - is an initial rotational velocity of a given rigid body

W1 and W2 - are vectors representing position of a wall ends

C - is a vector representing point of collision

X2 - is a vector representing position for a given rigid body at the moment of contact with the wall 
V2 - is a vector representing resulting velocity of a given rigid body after contact
R2 - is a rotational velocity of a given rigid body after contact

我知道我可能缺少一些参数，例如质量、阻尼、摩擦或描述墙 material 的属性，但我将其留给回答问题的人，因为可能有不同的模型。

我正在写 Dev Ramtal 和 Adrian Dobre 合着的一本书 "Apress Physics for JavaScript Games Animation and Simulations With HTML5 Canvas"，但是这个主题非常广泛、支离破碎和稀疏。所以很难得到一个简单明确的答案是如何解决这种冲突的。

Answer 1

我曾尝试为旋转圆盘与墙壁碰撞的情况编写一些伪代码。希望对您有所帮助：

# unit vector aligned with the wall
T = W2 - W1
T = T / norm(T)

# unit vector perpendicular to the wall, pointing towards the moving object and 
# hence perpendicular to T
N = [- T[1], T[0]]

# initial position and velocity:
X1 = [X1[0], X1[1]]
V1 = [V1[0], V2[1]]

R1  = angular velocity, positive if counter-clock-wise, negative otherwise

t_start = start time of simulation
t_stop = end time of simulation
k = friction coefficient, 
    that determines how much angular momentum 
    is converted into linear momentum during collision

t_col = t_start + dot(N, W1 + r*N - X1) / dot(N, V1)

if t_col < t_stop{
   V2 = V1 - 2*dot(v1, n)*n  - k*r*R1*T
   R2 = (1 - k)* R1 
   X_col = X1 + V1*(t_col - t_start)
   X_stop = X_col + V2*(t_stop - t_col)
   C = X_col - r*N
} else{
   X_stop = X1 + V1*(t_stop - t_start)
}

这是一个python版本，我只是想确保没有太多错误...

import numpy as np

# unit vector aligned with the wall
W1 = np.array([ -1, 0])
W2 = np.array([10, 1])
T = W2 - W1
T = T / np.linalg.norm(T)
# unit vector perpendicular to the wall, pointing towards the moving object and 
# hence perpendicular to T
N = np.array([- T[1], T[0]])
# initial position and velocity:
X1 = np.array([ 0, 5])
V1 = 0.3 * np.array([ 1, -2])
#angular velocity, positive if counter-clock-wise, negative otherwise
R1  = -1.2 

t_start = 0 #start time of simulation
t_stop = 3 #end time of simulation
k = 0.7 #friction coefficient, 
    #that determines how much angular momentum 
    #is converted into linear momentum during collision
r = 0.5 # radius of the disc 
# time of collision
t_col = t_start + np.dot(N, W1 + r*N - X1) / np.dot(N, V1)

t_start = 0 #start time of simulation
t_stop = 8 #end time of simulation
k = 0.7 #friction coefficient, 
    #that determines how much angular momentum 
    #is converted into linear momentum during collision
r = 0.5 

t_col = t_start + np.dot(N, W1 + r*N - X1) / np.dot(N, V1)

if t_col < t_stop:
   V2 = V1 - 2*np.dot(V1, N)*N  - k*r*R1*T
   R2 = (1 - k)* R1 
   X_col = X1 + V1*(t_col - t_start)
   X_stop = X_col + V2*(t_stop - t_col)
   C = X_col - r*N
else:
   X_stop = X1 + V1*(t_stop - t_start)

评估与墙壁碰撞后的刚体线速度和旋转速度

Evaluating rigid body linear and rotational velocities after collision with a wall

physics

kinematics

game-physics