So, the next question is obvious: is kinetic energy also conserved, just like momentum is conserved? The answer is: sometimes. For some collisions we call “elastic collisions,” both kinetic energy and momentum are conserved. Generally, elastic collisions occur between very elastic objects – like two rubber balls, or a pool ball. If we have elastic collisions in one dimension (meaning everything happens in a straight line), then we have two equations we can use: conservation of momentum and conservation of kinetic energy.
In addition to elasticity, there are two other types of collisions. When two objects collide and stick together, like a piece of clay hitting a piece, we call it a completely “inelastic” collision. In this case, momentum is still conserved, and we also know that the final velocity of the two bodies is the same because they stick together.
Finally, there are cases where two objects collide but don’t stick together and Don’t save kinetic energy. We just call these “collisions” because they are not one of the two special cases (elastic and inelastic). But keep in mind that in all of these cases, the momentum will remain the same as long as the collision occurs within a short time interval.
OK, now let’s consider a problem that is very much in the cradle of Newton. Suppose I have two metal balls of equal mass (m), ball A and ball B. Ball B starts at rest and ball A moves towards it at a certain speed. (let’s call it v1.)
Before the collision, the total momentum is mmv1 + meters0 = mmv1 (since ball B starts at rest). The total momentum after the collision is still mv1. This means that both balls can move at a speed of 0.5v1 or other combinations – as long as the total momentum is mv1.
But there is another limitation.Since it is an elastic collision, the kinetic energy must be return be protected. You can do the math (it’s not that hard), but it turns out that in order to preserve both KE and momentum, there are only two possible outcomes.The first is that ball A ends up with velocity v1 Ball B remains stationary. This is exactly what happens if ball A misses ball B. Another possible outcome is that ball A stops and then ball B has velocity v1. You may have seen this happen when a billiard ball hits a stationary ball. The moving ball stops and the other ball moves.
This is basically what happened to Newton’s cradle. If the collision between the balls is elastic (which is a fair approximation) and everything lines up (so it’s one-dimensional), the only solution for a ball on one side hitting the stack is to stop it and let The other ball moves instead. This is the only way to preserve kinetic energy and momentum at the same time. If you want to see all the details in this derivation, watch the video below:
What about inelastic collisions? It’s easy.Since both balls have the same mass and Same speed (because they stick together), the only solution is that they both move at 0.5v1 after the collision.In the case of a normal collision (neither elastic nor inelastic), both balls have velocities between 0 and v1.
As a demonstration, here are three colliding balls. The top shows elastic collisions, the bottom is inelastic, and the middle is in between.
I think this looks cool.
Video analysis of the ultra-fast cradle
There are a few things that make the crash in the slow-motion lad video different from the action of a normal Newton’s cradle. Instead of the five balls in the setup, the sixth, the one shot from the air cannon. This ball moves very fast – but it also looks slightly smaller than the other balls in the cradle, which means it has a different mass.
As you can see in the video, instead of simply bouncing outwards at the end of the post, four of the five balls completely snap off their strings and fly away as the base falls. This doesn’t work as a nice clicky office toy (it might punch a hole in your wall).