Let me be clear. I am not really an attacker. If someone wrote a report about ski jumping or something and misused the word “momentum”, no big deal. However, if you have a show that claims to be about SCIENCE and you are obviously putting a lot of money into this show AND a whole bunch of people will see and think this is science – then you need to be a little careful. I think shows like ESPN’s Sport Science are a good idea – you know, introduce some cool science ideas by using cool sports. This show just needs some help.
Yes, I know I make mistakes. I try to correct them when I become aware of them, but I am just one person. Also, sometimes I say things that aren’t quite true – but they are good lies. You know, like saying the weight is m*g (that isn’t always true). Ok – so back to my “commentary” on a particular Sport Science episode (at least they are short). Here is one about jumping in the snow ski half-pipe. Check it out (it’s not too long)
First, this episode has the same problem all the other episodes I have seen have – inconsistent use of terms. I don’t think I will talk about this today – it can be saved for a later post. Instead, I want to focus on two problems.
1 mph loses 3 feet
In talking about how important speed in the half pipe, Sport Science claims that for every 1 mph of speed a skier loses, he/she will lose 3 feet in height. I can’t get this one to work. First, I am not really sure about what they mean by speed. Is this the average speed going down and back up the half pipe or the speed at the instant the jumper leaves the pipe? I am going to assume it is the speed at the point the skier leaves – this will make things easier to analyze.
In this image (I found it searching for Simon Dumont, but I added my own stuff to it), I assume that this is like a half pipe jump. Since I am concerned with height, it is best to use the work-energy principle. If I consider the skier plus the Earth as the system, then while in the air there is no work done so that:
The key is that if I call the bottom zero potential energy, then at the bottom the skier only has kinetic energy and at the top, there is only potential. I can write this as:
So, here is a functional relationship between speed and height. Let me plot this from an initial speed of 1 m/s to 15 m/s (24 mph is about 10 m/s):
And here is the problem – this is not a linear relationship. Let me plot this in units of feet and mph – just so I can make it more easily agree with the video.
So, going from 24 mph to 45 25 mph gives a change in height of about 1.5 feet. However, going from 10 mph to 11 mph only has a difference of about 0.8 feet. Really, I want to know if I change the initial speed, what happens to the change in height – this could be a great set up for a calculus problem. What is the rate that h changes with change in v? This can be written as:
As is true for any parabola, the rate of change (slope) is proportional to the value of the velocity. If I use a speed of 24 mph, I get that the height changes about 1.8 feet per mph. According to this model, you would have to be going 44 mph to have a speed loss of 3 feet per mph. I am still not sure this is what the show was talking about.
Power – again
The other aspect I would like to examine is the power. Note that it seems they are using the term “power” in two different ways in this episode. I will use the definition:
And power for what? I am going to guess the only thing would be the power he produces while pushing off the lip of the half pipe? Not sure this really makes much sense, but it is the best I can do. Sport Science claims that Simon Dumont produces 24,000 watts. Seems high, but at least it is not as high as the power they claim from Marshawn Lynch – 57,000 watts – really.
There is a greater chance I can get this value to seem reasonable – mostly because if it is for a jump, it would be a very short time interval. I have no idea how Sport Science came up with 24,000. All they state is that they put motion sensors on him so that they could make an animated skeleton move like him.
To explore if this is a reasonable value, I am going to look at the power produced from Kobe Bryant as he appears to jump over a car – (note that I am pretty sure the video in that post is not quite real, but I do think Kobe can really jump like that). This is a good test case to look at because I know how high he jumped and I can look at the time it took to jump. This will let me calculate the power.
Here is the plan. I know the energy that Kobe produces because I know how high he went. The time can be determined from the video. This is the graph of Kobe’s height has a function of time – including the height of his estimated center of mass:
Looking at this graph, Kobe’s center of mass starts at 1.65 m and goes up to about 2.9 meters. If Kobe has a mass of 93 kg, then his change in energy is:
Looking back at the video, Kobe is in the process of jumping for 0.231 seconds. Note to other video analysts. You can not easily get the time for these events from something like Quicktime. Quicktime rounds the time of each frame up to the second – which is useless. I use Tracker Video Analysis for this. So, the power from Kobe is:
If Kobe produces just under 5000 watts – how is Simon supposed to produce 24,000 watts. I would love to hear from Sport Science and see how they get these power values. Please don’t tell me it is this:
Sport Science: “Hey look! Tiger Woods produces 40,000 watts in his golf swing!”