- 131 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about 'Collision Response' - lynelle

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

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 (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.

Download Presentation

Connecting to Server..