So here’s the obvious next question: Is kinetic energy also conserved, just like momentum is conserved? The answer is: sometimes. For some collisions which we call “elastic collisions”, both kinetic energy and momentum are conserved. In general, elastic collisions occur between very bouncing objects – such as two rubber balls or swimming pool balls colliding. If we have an elastic collision in one direction (that is, 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 lump of clay hitting a block, we call this a completely “inelastic” collision. In that case, momentum is still conserved and we also know that the final velocities of the two bodies are the same, since they stick together.
The last is the case where two objects collide but do not stick together and no conservation of kinetic energy. We just call these “collisions”, because they are not one of two special cases (elastic and inelastic). But keep in mind that in all these cases momentum is conserved as long as the collision is for a short period of time.
Okay, now let’s look at a very important problem in Newton’s cradle. Suppose I have two metal spheres of equal mass (m), sphere A and sphere B. Ball B starts without rest, and sphere A is moving towards it with some velocity. . (Let’s call it vfirst.)
Before the collision, the total momentum will be mvfirst + m × 0 = mvfirst (because ball B starts to rest). After the collision, the total momentum should still be mvfirst. This means that both balls can move with a velocity of 0.5vfirst or some other combination — as long as the total momentum is mvfirst.
But there is another limitation. Since it is an elastic collision, the kinetic energy must be But also preserved. You can do the math (it’s not too difficult), but it turns out that to preserve both KE and momentum, there are only two possible outcomes. First, ball A ends up with velocity v .first and sphere B remains at rest. This is exactly what would happen if ball A missed ball B. The other possible outcome is that ball A stops and ball B has velocity v.first. You may have seen this happen when a billiard ball hits the top of 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 balls is elastic (which is a reasonable approximation) and everything is aligned (so that it is one-way), then the only solution for a ball on one side hitting the stack is Let it stop and let another ball move instead. That is the only way to preserve both kinetic and momentum. If you want to know all the details in that derivation, this is the video for you:
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What about inelastic collisions? It’s quite easy. Since both spheres have the same mass and have the same velocity (because they stick together), the only solution is to both move at 0.5vfirst after collision. In the case of a simple collision (inelastic or inelastic) both balls will have some velocity between 0 and vfirst.
This is just a demonstration, here are three balls colliding. The upper part shows an elastic collision, the lower part is inelastic, and the middle is somewhere in between.
Video: Rhett Allin
I think that looks great.
Video analysis of super fast cribs
There are a few things that make the collision from the Slow Mo Guys video different from the normal Newton crib collision. Instead of five balls in the set-up, there is a sixth ball, which is fired from the cannon. This ball moves super fast – but it also looks a bit smaller than the other balls in the cradle, which means it has a different mass.
And as you can see in the video, instead of the ball at the end of the pole just bouncing outside, four of the five balls bounced off the rope completely and flew away when the base fell. This won’t work as a nice office toy (and it could make a hole in your wall).
