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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 l.jpg

Collision Response

CS 134

Soon Tee Teoh


Collision response2 l.jpg
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 l.jpg
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 l.jpg
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.


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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 l.jpg
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


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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


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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


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Perfectly (continued)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.


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