Collisions of more than two bodies

In summary, the formulae for the velocities of two bodies after a perfectly elastic or inelastic collision in 2D can be derived from three equations. However, when dealing with collisions of three or more bodies, the number of unknowns increases while the number of constraints remains the same. This can be problematic and material properties may play a significant role in the outcome. It has been observed that the dynamics of Newton's cradle can be affected by the shape of the bodies used, and treating it as a series of two-body collisions may not always be accurate. This approach may work with spheres, but not necessarily with cylinders.
  • #1
greypilgrim
533
36
Hi.

The formulae for the velocities of two bodies after a perfectly elastic or inelastic bodies, let's say in 2D, (e.g. billiard) can be derived from three equations: conservation of energy and conservation of momentum in two dimensions.

But how do you treat collisions of three or more bodies? With each additional body the number of unknowns rises by two (velocity in x and y direction), but the number of constraints is still three.

Is the problem stable with respect to shifting the bodies by arbitrarily small (or virtual) displacements such that there are only two-body collisions to consider?
 
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  • #2
greypilgrim said:
Is the problem stable with respect to shifting the bodies by arbitrarily small (or virtual) displacements such that there are only two-body collisions to consider?
It is not, and in general material properties will be highly relevant for the outcome if the overall process cannot be described as independent two-body collisions separated in time.
 
  • #3
I once read that the dynamics of Newton's cradle is higly dependant on the shape of the bodies used, using cylinders instead of spheres will not show the same behaviour.

Elsewhere I read that one may treat Newton's cradle as if the spheres are separated by a small distance such that only two-body collisions occur. With this assumptions one can easily derive the observed dynamics by just using the equations for two-body collisions. So is this a coincidence and wouldn't work with cylinders?
 
  • #4
Apparently not. It would work if the cylinders are actually separated a bit.
 

Related to Collisions of more than two bodies

1. What is a collision of more than two bodies?

A collision of more than two bodies is a situation in which three or more objects come into contact with each other and interact with one another, resulting in changes to their motion and energy.

2. How is the motion of each object affected in a collision of more than two bodies?

The motion of each object is affected by the forces exerted on them during the collision. These forces can cause changes in velocity, direction, and rotation of the objects involved.

3. What is the conservation of momentum in a collision of more than two bodies?

The conservation of momentum states that the total momentum of a closed system (in this case, the objects involved in the collision) will remain constant before and after the collision. This means that the total momentum of the objects before the collision will equal the total momentum after the collision.

4. How do you calculate the final velocities of objects after a collision of more than two bodies?

To calculate the final velocities of objects after a collision of more than two bodies, you need to use the conservation of momentum and energy equations. These equations take into account the masses, initial velocities, and the type of collision (elastic or inelastic) to determine the final velocities of each object.

5. What are some real-life examples of collisions of more than two bodies?

Some common examples of collisions of more than two bodies include car crashes, billiard balls colliding on a pool table, and planets interacting with each other in space. These collisions can also occur in sports, such as a football tackle or a soccer player colliding with another player on the field.

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