# DMD:Background:The DMD Model Equations

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The relationship between time and position gives:

where is the vector position of the particle and 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, , we have ,

Where

The relation leads to a quadratic equation for the *ij* collision time, .

Where :

discriminant. Note that in m/s is a large number while is very small.

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At the time of the collision, the velocities of the particles change according to,

We can derive this formula by assuming that particle |

Note that and must be updated to the point of collision before computing the velocity changes.