Elastic collisions of smooth spherical objects: Finding final velocities in four simple steps

Collisions of two spherical objects are a common topic in undergraduate lectures. The final velocities for elastic collisions along a straight line are easily determined from the impact velocities, but the determination of the final velocities for collisions in a plane and in space requires additional information and more demanding algebraic manipulation. In this paper, we derive a method for finding final velocities after elastic in-plane and in-space collisions of smooth spherical objects of arbitrary masses, radii, and impact velocities directly in the laboratory frame without intermediate transformations between frames with different velocities. The forces between smooth spherical objects during the collision are central. Therefore, the velocities of colliding objects can change only in the direction of the forces. This allows for reduction of the in-plane and in-space elastic collisions to collisions along a straight line parallel to the direction of the forces between objects, that is, along the vector connecting the centers of both objects during the collision. Components of velocities perpendicular to the forces do not change during the collision. The methods for finding final velocities after transformation to the center-of-mass frame or to the relative-object frame are compared with the four-step method in the laboratory frame for two examples: for a general collision and for a collision of billiard balls. Applying the reasoning of the new method provides additional insights into the physics of collisions analysed in alternative frames.