The relationship between time and position gives:
(1) rij=ri−rj=(ri0+ui(t−t0))−(rj0+uj(t−t0))
=(rijx,rijy,rijz)
where ri is the vector (rix,riy,riz) position of the ith particle and $ u_i$ is the vector velocity.
Note: html to make bold typeface for vectors in math mode is unknown to this author. Sorry.
At the time of contact, tijc, we have ,
(2) σij2=(rijc)2={(ri0+uitijc)−(rj0+ujtijc)}2
≡{(rij+uijtijc)}2={rij∙rij+2tijcrij∙uij+uij(tijc)2}
Where rij=(rij∙rij)1/2={(rijx)2+(rijy)2+(rijz)2}1/2
The relation leads to a quadratic equation for the ij collision time, tijc≡(tij−t0).
(3) uij2(tijc)2+2bijtijc+rij2−σij2=0
(4) tijc={−bij−(bij2−uij2(rij2−σij2))1/2}/uij2≡{−bij−Dij1/2}/uij2
Where :
(5) bij=rij∙uij
(6) uij2=(ui−uj)2
(7) Dij=(bij2−uij2(rij2−σij2))≡ discriminant. Note that uij in m/s is a large number while rij is very small.
Separating | Approaching, but miss | Colliding |
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) 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xtccpuW52bhXAbLhafy8dX4vV9s+Q0001E3I0000A0000A001gbizWL29hbOZzNxKlGsoaWVgT+SAqqoBZmZiFVVBZmIABJA0MNpLLNZvrem39mY9bGZuoLMysaVCNlNq6y8cpDGSOxBZeT7KgPZyqgsKK3Z6j723LZm+1ys+2cQ/ZY6VJIxaeJgARPYPcq/A5BgMZjLMA8hVZNRy/lMluLN3Nz5mH7fI5Lxs1USeRaUaoAlZG5PKr8iSOFaR5XAXv1HxqxCklqz5JzBzrcVqsqFi+mCf6lvLt5Ltr664MPIY6PKQivm7mWzldW7rXy+TsX4VfggOI53dQwBI7AcgEjngnUMpUxQ9bEqYq5k8TBHtk+NMbkJ6gjU2uSieJ16oWJYoOFLEsR2JOp/qE/79/+2P8A2tTlFeRoOHcUvZTqzlWk30S16nnw9S6No+q+8cBZ/wBIZ593Y1uxcLXrw5CNiBwY2XxQOgI90ZVb5M3kPVYzeu1dy4HdWL/U9vZWtkaobxyNC/yhk6hjHIh+UcgDLyjgMvPBAOuWNbLZu4rGzN319x143mqyKtPKV17HyVmkUmYKnJeSH5uo6sSrSooDShhCdJYyjs3LnOlb40ba+eYyeFLZrv4NZ8dMbt426r0140LdW/Rr3qNmG1UsxLNBPDIHjlRhyrqw9mUgggj2IOvbVc+qjTTTQDTTTQFGfQplKGQ+m3B1Kc/lmxlm3VuL0YeOUzvMF5I4P9nNG3I5Hy4/IIF56pP6VbdV73q1RSzC1uH1Eyk0sAkBkRHKBHK/kKxjcAn2JRuPwdXZoBpppoBpppoBpppoBqlPqgyUr2NqbYXusFqaxlJyHPWVaojRImT8MPJajlBJ+LQL7ckFbr1zv9QslyT1cpQ2w4p18CrY4snVTJJYkFvq3HzIEVLsOT05T+nue06azJGi5nryocJrzjvjGnq8fchWmmmrZ5/GoT/v3/7Y/wDa1NteP2dT7/8AUPtYPvPF4fuPGPJ4+eenb89effj8c6w1kt2lyqHXlZ6otfU9tNNNZKhdX0x5KWz6f3MLJ3YYDKzUIpGcntEyR2Y0Vf8AYWNLCwqoJAWIccAhRaeqO+l+S4M3vOrGH/S+aM4ITlPvWSVJh34/q8MdPlOfYFG4Hflrx1TksSZ6J4JXlccOoVJbuKznzxq/caaaaibQa02+aWdyG0MrT2vlv0jOSVn/AE+4URljnA5TuHRx0LABviT1J44PBG500BwB9O03qhu/6tbMG5NwZqtfq2TY3UMfcjqrL9iPFHHKkXEcqeXxxEKDyrsR+S2u/wDUZxGxdt4v1Dzm/KdLrnM3Wr1rkzkN8YQQOhI5TsojDAHq3hjPHIJMm0A0000A0000A0000A1R31QU5Y83szNsyfbg3sV1BPfyzJFOjcfjqFpSgnnnlk9iCSLx1pt67bx27ts29v5UzLWs9GEkL9ZIpI3WSORSQR2V0RgGBUleGDAkHMXh5KPE7P520qW+cdSa9/D+TlvTXyYMjj7tvDZuOGPMYyVa2QSA8xeXxo4aM/ujI6OvPyCuAwVgyj61dTzqedri3qW1WVKqsSi8MaaaaHCNNNZOBxF/c256G2cXJ4rNrmexNyoNenG8YnlUsCC4Eiqg6t83QsvQOQbwsstWVnVva8Leisyk8L89lu/Qtv6XKcseys1lmZPBls9YmgUE9lWFIqb9h+ATJVkI4J+JU+xJAtnWFgsXQweDoYTFwfb0MfWjq1Yu7N44o1CovLEk8KAOSSf51m6pN5eT0XaW6tqEKEdopL6LA0001gsDTTTQDTTTQDTTTQDTTTQDTTTQDTTWPauRV7VSu6uXtSNGhAHAIRn9/wDop0BE/Un05wu81F5mfH52CDxVMlCW+IB7KssYYLPGCW+L+6h3KNGzdxQm7Nv7k2fZmh3Bh7slOLsVy9Kq8tOWNQC0rlO5rKFILebqqnt1eRUL66v01OM3HY0PGOXLLiyzWWJfuWj/ALXf2OMcZnsHlJ2r4zM467Mq92jr2kkYLyBzwpJ45I9/8dfF/cm3aFt6l7PYqrYj47xTXI0deRyOQTyPYg/9ddN+q9/ZsMeAxO9NuV89Xy2Whp04bFKGzFFYflFkZZTwOAxHYAngn299SnCYnFYPFw4vCYyljKEHbxVacCwxR9mLHqigAcsSTwPySdcnxn5HVV/jqj15+O8dln65+xzTtHZW793WfFRxV3A0l7eXJ5nHSRKjAA9ErO0c0jHlfl8Y+Ox7ll8bX9sHY+A2VSmhw8M72LXQ3LtqYy2LLKCAWY+yryWYRoFjUu/VV7HUl01xSm5bnbuE8AsuFRaoR1e8nq/r+MDTTTUTdDTUawG9MXmt77g2jVr3EvYGOvJakkRRE4m79ehDEnjxnnkD9vzqS6AaaaaAaaaxq92KbIWqSq4krBC5IHB7gkcf/mgMnTTTQH//2Q==) 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) 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bi>0 (forget about it) | Dj<0 (forget about it) | Dj>0 (schedule it) |
At the time of the collision, the velocities of the particles change according to,
(8) mj△ui=(mi+mj)−2(bijc/σ2)rijc=mi−△uj
We can derive this formula by assuming that particle j is stationary (reference frame) and particle i is moving on the x-axis with equal mass. The j-direction after collision is given by the line of action rijc, since that is all j feels about momentum change. Conservation of momentum means that uif+ujf=uii with the geometric interpretation of a sum of vectors in the form of a triangle. Conservation of energy gives (uif)2+(ujf)2=(uii)2. The Pythagorean theorem applied to the conservation of energy means that this triangle must be a right triangle. Therefore, we can rotate the coordinate system such that y=uif;x=uif;uif=uiicosθ;ujf=uiisineθ
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)
Note that rijc and bijc must be updated to the point of collision before computing the velocity changes.