Sphere and disk collisions can be expressed more compactly and computed more conveniently in vector notation. Primarily, this involves converting the procedures of Example 7.10 to use the dot product of the relative position and relative velocity. (You may find useful information in the DMD module at Etomica.org.)

(a) Write a vector formula for computing the center to center distance between two disks given their velocities, u, and their positions, r₀, at a given time, t₀.

(b) Write a vector formula for computing the distance of each disk from each wall. Use unit vectors x=(1, 0) and y=(0, 1) to isolate vector components.

(c) Noting that energy and momentum must be conserved during a collision, write a vector formula for the changes in velocity of two disks after collision. (1) ab=abcosθ. (2) A unit vector with direction of a is: a/a.

(d) Write a vector formula for the change in velocity of a disk colliding with a wall.

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Example 7.10 Computing molecular collisions in 2D Let the diameters of two disks, o, be 0.4 nm, the masses be 16 g/mole, and the length of the square box, L, be 5mm. Start the disks at [1.67 1.67], [3.33 3.33] and initial velocities (nm/ps): [0.167 0.222], [-0.167 -0.222] where 1 nm = 109 m and 1ps = 10-¹2 s. Note that the gas con- stant 8.314 J/mol-K = 8.314 kg-nm²/(ns²-mol-K) = 8.314(106) kg-nm²/(ps²-mol-K). (a) Compute the temperature (K). (b) Compute the collision times with the walls. (c) Compute the collision times with the disks. Which event occurs first? (d) Compute the velocity vectors (m/s) after the first collision event.