In summary, this article focuses on an alternate approach to solving collision problems by determining the equal and opposite collision impulse for elastic collisions. The reduced mass and relative velocity are also taken into consideration. The aim is to develop a method for solving 2-dimensional elastic collision problems using a Cartesian plane. The article also discusses the possibility of adapting this theory for Compton scattering.
neilparker62
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Introduction
This article follows on from the previous on an alternate approach to solving collision problems. In that article, we determined the equal and opposite collision impulse to have magnitude ##\mu \Delta v## for perfectly inelastic collisions, ##\mu(1+e) \Delta v## for semi-elastic collisions and ##2\mu \Delta v## for elastic collisions which will be the focus here. Reduced mass ##\mu=\frac{m_1m_2}{m_1+m_2}## – where ##m_1## and ##m_2## are the colliding masses – and ##\Delta v## is their relative velocity along the line of collision. e is the coefficient of restitution.
Since the previous article focused on 1-dimensional collisions, the aim here is to develop a method of solving 2-dimensional elastic collision problems using a Cartesian plane in which the x and y axes are defined to be respectively parallel and perpendicular (normal) to the line of collision. The latter is defined by the post-collision direction of the stationary mass since it cannot attain momentum...

Last edited:
PhDeezNutz, Hrishikesh Edke and Greg Bernhardt
If you are looking at the elastic collision of two bodies, shouldn't you be solving for the transient elastic deformation of the bodies during the collision using Theory of Elasticity, involving Young's modulus, density, and Poisson ratio of the two bodies?

Thanks for the comment / query but I'm really not doing much more than follow the basic theory of 1-dimensional elastic collisions in which transient deformation is not usually dealt with. For example a bouncing ball deforms when it hits the floor but the energy is lost and then immediately regained. Very specifically in this article we are dealing with fully elastic 2D collisions (coefficient of restitution = 1). Have requested the graphic being used gets changed as it may create the wrong impression - car collisions are not at all elastic! Gravitational slingshots might be more pertinent examples of what's being described here.

Graphic has been changed - thanks Greg!

Great article, never thought about the relationship between refraction and momentum. Another amazing relationship is to apply a wave function to one of the particle sizes. Let one of the particles expand and contract at a certain frequency which adds to the dynamics of the collision (ie. if expanding or contracting, the particle will have another aspect to the collision). This extends the refraction/momentum model to include diffraction around edges.

Thanks for your kind comment - what I had been trying to do is see if somehow the theory can be adapted for Compton scattering. But haven't made much progress as yet. Can't quite figure what ##2 \mu \Delta v## (elastic collision impulse) looks like when we're talking about a photon/electron collision.

## 1. How does this alternate approach differ from traditional methods of solving 2-dimensional elastic collisions?

This approach takes into account the individual masses and velocities of the colliding objects, rather than assuming equal masses or using simplified equations. It also considers the conservation of momentum and energy separately, rather than combining them into one equation.

## 2. What are the benefits of using this alternate approach?

This approach provides more accurate results and can be applied to a wider range of collision scenarios. It also allows for a better understanding of the underlying physics behind 2-dimensional elastic collisions.

## 3. How does this approach account for non-ideal conditions, such as friction or air resistance?

This approach can be adapted to include external forces, such as friction or air resistance, by adding them as additional terms in the equations for momentum and energy conservation.

## 4. Can this approach be used for 3-dimensional elastic collisions?

Yes, this approach can be extended to 3-dimensional collisions by adding an additional dimension to the equations for momentum and energy conservation.

## 5. Are there any limitations to this alternate approach?

This approach assumes that the colliding objects are perfectly elastic and do not deform upon impact. It also does not take into account rotational motion or other factors that may affect the collision. Therefore, it may not be suitable for all types of collisions.

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