Applying conservation of momentum in elastic and inelastic collisions is a standard fare in both algebra and calculus-based introductory physics classes. In the case of elastic collisions, it is not uncommon to see treatments limited to cases where one of the objects is initially at rest. The reason for this is of course that the algebra involved in conserving both momentum and kinetic energy is difficult because the latter is quadratic in the final velocities. In this note, I develop a very compact and quite general treatment of elastic collisions by working not with final velocities but rather the magnitude of the momentum exchange involved.

Consider a collision between two smooth, non-deformable, non-rotating disks (or spheres) as sketched in Fig. 1. The figure is drawn in two dimensions, but the derivation that follows works also in three dimensions. During the collision, the disks exert equal but opposite impulses on each other which are directed along the line joining their centers. (If the disks are smooth, they can exert no tangential forces on each other, hence no rotations can be induced.) Let $Δp1$ be the change in momentum of disk m1. If p1 and p2 are the pre-collision momenta, then the final (post-collision) momenta will be

$p1f=p1+Δp1$
(1)

and

$p2f=p2−Δp1.$
(2)
Fig. 1.

Elastic collision between two non-deformable, non-rotating disks. Equal and opposite impulses occur along the line joining the centers of the two disks.

Fig. 1.

Elastic collision between two non-deformable, non-rotating disks. Equal and opposite impulses occur along the line joining the centers of the two disks.

Close modal

If the disks are non-deformable, no energy can be lost due to any work which causes changes in shapes. The pre-collision and post-collision kinetic energies are then equal, that is,

$p1f•p1fm1+p2f•p2fm2=p1•p1m1+p2•p2m2.$
(3)

Substitute Eqs. (1) and (2) into (3) and carry out the scalar products. After some cancellations, what remains is

$2(m1p2−m2p1)•Δp1=(m1+m2)(Δp1)2.$
(4)

Now let $ê2→1$ be a unit vector from the center-of-mass of m2 to the center-of-mass of m1. We can then write

$Δp1=| Δp1 |ê2→1.$
(5)

With this, Eq. (4) can be simplified to

$| Δp1 |=2(m1p2−m2p1)•ê2→1(m1+m2)=(2m1m2m1+m2)(v2−v1)•ê2→1.$
(6)

Equation (6) is the main result of this note. It is easy to show that it predicts the correct final velocities for the standard textbook cases of strike-from-behind and head-on collisions, and also that the two masses will scatter at 90° to each other in cases of equal-mass non head-on collisions where one is initially at rest; readers are encouraged to check these cases for themselves.

An informal survey of both introductory and more advanced texts in my college's library reveals no treatment of elastic collisions along this line. This derivation should be entirely suitable for calculus-level students and requires no introduction of concepts such as center-of-mass-frame velocities. Its power and simplicity opens up the possibility of taking on examples that are beyond the usual strictly algebraic treatments.

The author is grateful for the comments of three anonymous reviewers, which helped to improve this note.