Momentum and Collisions

The air track. Air blows out of tiny holes in the track, so the gliders float on a thin cushion of air with almost no friction. That matters: with no outside force pushing along the track, the gliders only push on each other. Velocities to the right are positive and to the left negative. Each glider has a spring bumper; set the bounciness to 0 and they get Velcro pads instead, so they stick.

The center of mass. The white triangle under the track marks the balance point of the two gliders. Watch it during a crash: it glides at the same steady speed before, during and after, whatever the gliders do. That is momentum conservation you can see.

The bars. Momentum bars point up for rightward momentum and down for leftward. The white “total” bar never changes. The kinetic energy bars are different: in a sticky or squashy crash, part of the energy disappears into heat, sound and bent bumpers, shown as the red striped part.

The force graph. In slow motion you can watch the push between the gliders rise and fall during the 60 milliseconds they touch. The area under that curve is the impulse: exactly the momentum that moves from one glider to the other.

A car that hits a wall at 50 km/h (31 mph) has to lose all its momentum. If its front were rigid and crushed only 10 cm, the stop would take 14 ms, and everything inside would slow down at about 100 times the strength of gravity (100 g). Modern cars have crumple zones: the front is built to fold up in a controlled way, so the cabin stops over a much longer distance. In this model, a 0.6 m crumple makes the stop last 86 ms and brings the average down to 16 g.

A seat belt makes you ride down with the car, using its crumple zone, instead of hitting the dashboard or windshield at full speed a moment later. The belt also stretches a little, and an airbag adds a soft, wide cushion. Each adds stopping time and spreads the force over strong parts of the body. The U.S. National Highway Traffic Safety Administration says that for front-seat passenger-car occupants, buckling up cuts the risk of being killed in a crash by 45%, and that seat belts saved an estimated 14,955 lives in 2017. It also says airbags are designed to work with seat belts, not replace them.

On a real table, pool balls lose a little energy in each hit, and the white ball is usually rolling, so its spin bends its path a moment after the hit. The 90° rule works best for a ball that is sliding without spin when it hits. Lower the bounciness slider and the angle closes to less than 90°.

A rocket in space has nothing to push against, yet it speeds up. It works by throwing hot gas out of its nozzle very fast. The gas gets momentum backward, so the rocket must get the same amount forward: the total stays what it was. A skater who throws a heavy ball while standing on ice glides backward the same way.

As the fuel burns, the rocket gets lighter, so each kilogram of exhaust speeds it up more. Adding it all up gives the rocket equation, Δv = u ln(m₀/m), where u is the exhaust speed and m₀/m is the starting mass divided by the final mass. With exhaust leaving at 3 km/s and two thirds of the starting mass as fuel, Δv = 3 × ln 3 = 3.3 km/s. Build and fly your own in the Rocket Builder.

René Descartes argued in the 1640s that the total “quantity of motion” in the world stays the same, but he measured it as size times speed, ignoring direction, and that doesn’t work. In 1668 the Royal Society in London asked for papers on how bodies collide, and John Wallis, Christopher Wren and Christiaan Huygens each sent in rules. Newton credits all three in the Principia.

Newton tested the rules himself, by hanging balls as pendulums, letting them swing into each other, and judging their speeds from how far they swung. For balls of tightly wound wool, he found they bounced apart at about 5/9 of the speed they met at; for glass balls, about 15/16. That ratio is the bounciness e on your slider.

Cars. Crash tests, crumple zones, seat belts, airbags and the water-filled barrels in front of highway pillars are all impulse at work. The cars course shows how the rest of a car works.

Sports. Catching a fast ball, you pull your hands back to make the stop last longer. A gymnast lands on a thick mat for the same reason. A bat or racket does the opposite: its large mass and speed hand a lot of momentum to the ball in a very short time.

Space. Rockets, and the small thrusters that turn spacecraft, all work by throwing mass one way to go the other. See Orbit Lab for what happens next.

Pool and bowling. Every break shot and every strike is a chain of collisions, each one passing momentum along. Players who aim cut shots are using the 90° rule from the table above.

The air track. Each glider moves at a constant velocity until the bumpers touch. The contact is a smooth push that rises and falls like half a sine wave over 60 ms, with its area set to the impulse that the momentum and bounciness formulas require. The same force acts on both gliders in opposite directions, so the total momentum is the same at every instant. The motion during contact has an exact formula, so nothing is approximated step by step. Real bumpers don’t push in a perfect half sine, but any shape with the same area gives the same final velocities.

The crash stops. Each stop uses the same half-sine force, lasting Δt = 2d/v for a stopping distance d; its peak is π/2 times the average. The distances (2 mm for a floor, 10 cm for a rigid car front) are round numbers chosen to show the idea, not measurements. The egg cracks when the average force tops 25 N, a rough line, since real eggs vary a lot.

The pool table. Balls are 57 mm smooth discs of equal mass. Collisions are instant and along the line joining the centers, so no spin is passed on. A gentle steady slowdown stands in for rolling resistance, and the cushions return 80% of the speed.