Conservation of Energy: Some Ups and Downs


Ruan Van der Merwe

The other day whilst browsing one of the favourite “professional” social media sites I came across an interesting post. The title read something along the lines of “The reason we have ups and downs in life is what allows us to excel (or accel-erate (the –erate is silent)… but we will get to that). Along with this motivational quote was a video…

In the video we have two metal balls (same size and same mass) that are held on two rails and let go (from the same height). However, the one rail is a parabolic shape where the other is a wave shape (sinusoidal). The first question that popped into my mind was “which ball will reach the end of its rail first?”. Here is where I made my first mistake; my guess was that they will get to the other side at the same time (if you also thought this, you are not alone). I gave some of my colleagues the same “test” and they also had the same intuition as I did. Here’s what we thought:

The balls are released at the same time AND from the same height so we have the same energy in the system, right? So where the ball on the parabolic rail will not go through undulations the other would gain and lose kinetic energy as it goes up and down. The conundrum now is whether the ball in the wave track ‘gains’ and ‘loses’ the same amount of kinetic energy meaning, the average velocity (scaled by the extra bit of track) is the same.

If you already had a look at the video, I couldn’t wait either, you would have seen that the ball on the wave rail is the victor. To my embarrassment, I thought the video is some first class example of video editing. I, therefore, had to investigate so I implored some math and Simcenter Motion.

Here is an explanation of what is happening…

I started with the basics, Conservation of Energy (with some assumptions):

I decided on two profiles and mapped them out:

rail profiles

Now I can use these as input for the conservation equation which results in the next figures…

From these figures, we can at least see that the conservation of energy calculations was implemented correctly (I needed this assurance after my blooper of guessing the outcome wrong). Cool, but let’s compare apples with apples. Here is the running-average velocity graph for the two cases:

Mmm, here we start seeing something interesting. The average velocity of the ball going through the undulations is larger than that of the ball on the parabolic shape.

We can also compare these results to what has been simulated in Simcenter Motion:

And obviously an animation:

So what we see is that the total energy of the balls remains constant and equal throughout their journey over the rails, but the proportion of energy stored in kinetic form is on average higher for the ball on the undulated rail. This counteracts the fact that the ball has to travel a greater total distance and leads to it reaching the end of the rail earlier.

Going through this blog you probably expected this result but this is the part we (Yes, I am taking everyone down with me) missed. It is totally obvious now but the epiphany took longer than it should have. The question on my mind at this stage is “At which point does the velocity gain even out with regards to the length of the track?”. This could be a nice optimisation sequence that can be run with either something like a DoE (Design of Experiments) or using the Monte Carlo method. Maybe I should stop making things too complex and just set up a track that can be repeated incrementally and see when the two balls match.