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Collision Response. CS 134 Soon Tee Teoh. Collision Response. Sometimes, it is necessary to calculate what happens after two objects collide. Where do the objects move? Example 1: What is the velocity of a billiard ball after it collides against the edge?
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Collision Response CS 134 Soon Tee Teoh
Collision Response • Sometimes, it is necessary to calculate what happens after two objects collide. • Where do the objects move? • Example 1: What is the velocity of a billiard ball after it collides against the edge? • Example 2: What are the velocities of two billiard balls after they collide? • We will consider these two cases, which serve as generalizations of collisions.
Movable object collides with Immovable • Example: Billiard ball against edge of table • What happens is like simple reflection. • Let the unit vector in the direction that the object hits the rigid surface be V. • Let the unit normal of the surface be N. • Then, the vector after collision R = V + 2N((-V).N) N q q V R surface
Elastic and Inelastic Collisions • Collision involves two steps. • 1. Period of compression: Two objects collide and deform • 2. Period of restitution: After compression, they bounce off each other. • If during restitution, entire deformation recedes and none of the mechanical energy is lost (changed to heat for example), this is called elastic collision. • If there is no restitution (that is, the two objects remain stuck to each other after collision), this is called inelastic collision. • Real-life situations are in between.
Collision of a Sphere with another Sphere: They both move • Suppose that two spheres collide. • As we model objects as spheres, a sphere is just a generalization of an object. • We assume elastic collision. Elastic collision means that no energy is lost during the collision. Therefore, both momentum and kinetic energy are preserved. • The velocity of the spheres after collision depends on their initial velocities, angle of impact, and mass. • In elastic collision, these two objects will bounce off each other and continue moving.
How to calculate sphere-sphere elastic collision response • Let v0 and v1 be the velocity vectors of the two spheres S0 and S1 respectively. • Let vc be the vector from the center of S0 to the center of S1. • Let v0c be the projection of v0 onto vc, and let v1c be the projection of v1 onto vc. • Let vp be the vector perpendicular to vc. Note that vp lies on the plane containing v0 and v1. • Let v0p be the projection of v0 onto vp, and let v1p be the projection of v1 onto vp. • Let M0 and M1 be the masses of S0 and S1 respectively. v1 v1p vp v1 v0 vp v0p v0 vc v0c v1c
How to calculate sphere-sphere elastic collision response (continued) • Let the new velocities after collision be r0 and r1 respectively for S0 and S1. • Then, r0 and r1 are given by the following formulas: r0c = v0c*(m0-m1)/(m0+m1) + v1c*2m1/(m0+m1) r0p = v0p r1c = v0c*2*m0/(m0+m1) + v1c*(m1-m0)/(m0+m1) r1p = v1p r0 = r0c + r0p r1 = r1c + r1p
How to calculate sphere-sphere inelastic collision response (continued) • But, how to find v0c, v0p, v1c and v1p in the first place? • The unit vector joining the center of the spheres, vc = (centerS1 – centerS0)/ ( | centerS1 – centerS0 |) • Then, multiply dot product v0c = vc * (vc.v0) v0p = v0 – v0c v1c = -vc * (-vc.v1) v1p = v1 – v1c
Perfectly inelastic collision • Now, suppose we have inelastic collision. • Happens when two pieces of clay collide and stick together. • Momentum is conserved, but kinetic energy is not conserved. • From conservation of momentum, u = (m1v1 + m2v2)/(m1+m2) • where u is the velocity of the (combined) object after collision, m1 and v1 are the mass and velocity respectively of object 1 before collision, and m2 and v2 are the mass and velocity of object 2 before collision.
