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