Gas molecules whiz around. When they hit our skin, the faster molecules feel "hotter". When molecules bounce against each other the hotter molecule gets cooler and the cooler gets hotter.
      The speeds converge toward the average.
A hot cup of coffee cools because its molecules bounce off the cooler molecules of the cup and air.

Is convergence of speeds inevitable? Yes. The proof follows.

In this note, molecule collisions are modeled as bouncing balls. Those above range in speed from cool blue to red hot. All balls have the same mass, as do all molecules of any given kind.


Try adding balls above by pointing the mouse into the topside color injector and holding down the button. When most of the balls are green, try adding a bunch of hot (red) balls. What is the eventual color?


Let's start with the simplest case, a head-on collision. The diagram shows two balls about to collide. The length of the arrows shows the ball's relative speed. Ball A's arrow is twice the length B's. A travels twice as fast as B so they will collide at the thin blue cross.

Adjust the speed of ball B by dragging it.

The thin blue cross signifies a coordinate space centered at the point of collision. The x-axis passes through the centers of the two balls. and the y-axis splits the angle between them.


We can show names and speeds for the arrows. Here the speed arrow of ball A is named VA and its value is 140. Drag ball B to set its speed; it is now ‌‌70.

The diagram muddles the concepts of vectors and ball paths. A velocity vector is solely a direction and speed; it is NOT rooted in space as are the arrows.

Speed is in pixels per second. At a speed of , a ball would cross this current window in one second.


When two balls hit head-on, they exchange momentum. The ball on the right continues the momentum from the one on the left, and vice versa.

Here ball B's velocity is zero; it sits at the point of collision. Click ball A. It hits ball B and stops, thus taking on the previous momentum of B. The exchange gives B the momentum A had, so moves on to the right.

Momentum depends on both velocity and mass. We are imagining balls of identical mass, so velocity and momentum are proportional. These results carry over to molecules where all molecules of any given kind have the same mass.


Even if both balls are moving, they swap momentums. Drag ball B to different initial momentums and click ball A to bounce the balls. After a collision ball B reverses direction and takes on A's former momentum. And ball A similarly takes on B's former momentum.

Exchange of momentum is often demonstrated with Newton's cradle, as discussed at minute 15 of this video tutorial by Louis A. Bloomfield, Univ. of Virginia


Moving on, we focus on two balls, A and B, colliding at an angle. The upper (bluer) arrows show their initial relative speeds and directions. The lower (greener) arrows are their speeds and directions after the collision.

The arrows are velocity vectors named vA and vB on the ingoing side and v 'A and v 'B on the outgoing side.

The diagram is in two dimensions, but note that for any two colliding balls there is always a 2-D plane in which the collision occurs.


Click ball A. The balls collide. Try other speeds by dragging B and clicking A again. The initial speeds of the balls are vA and vB (blue arrows.)

After a collision, the speeds are v 'A and v 'B (green arrows). At present the four speeds are

The values converge. That is, after the collision the resulting speeds (green) are closer to the average than were the initial speeds (blue).


A vector can be "projected" into a pair of vectors, one on each axis. In the diagram, vector vA projects to xA and yA, and similarly for the other four vectors. The traversal of vA is equivalent to traversing xA and yA, simultaneously.

Here you can see that ball paths are NOT the same as vectors. A ball path of xA and yA would take longer than vA. The equivalence only holds when xA and yA act on the ball simultaneously.


With the x and y projections we can easily see the relation of the incoming vectors to those outgoing. Nothing about the bounce affects the vertical progress of the balls. So the incoming and outgoing speeds are the same. That is, (as shown) y 'A is the same as yA and the same for y 'B


As we saw with the one-dimensional bounce, the two balls exchange their momentums when they collide. The outgoing B has the same momentum as the ingoing A. And the outgoing A has the same momentum as the incoming B.


Our careful choice of coordinate space has the y-axis splitting the angle between the incoming balls at the point of impact. This means that the two angles marked "a" are equal. Since the two incoming vectors are right-angle triangles with one equal angle; they are congruent triangles.


Now we reorient the four speed vector triangles. Because the inbound triangles are congruent, they nest nicely. As shown. The shorter sides of the triangles are the x and y vectors from the decomposition. The longer sides — the hypotenuses — are the speed vectors themselves.

By construction, the outbound (lower) triangles share the length and direction of various inbound edges. Hover the mouse over an outbound triangle to see where its edges match those of the inbound triangles.

Drag the "drag me" ball to adjust the triangle sizes.


Our goal has been to prove that the speeds converge; that is, that the magnitudes of the outbound momentums v 'A and v 'B are between the inbound momentums, vA and vB.

Click on either outbound triangle. Its hypotenuse glides up to lie between its ends in the inbound nested rectangles. In both cases the hypotenuse is clearly of a length between the two incoming lengths. The outbound speeds are closer to the average.
      THUS: Speeds converge when balls bounce.
Molecules are not quite the same as balls. but their speeds converge similarly.