You know Ninja Warrior: the Japanese obstacle course game, for ninjas. There are several events that come up quite often that look super difficult. One of these is the salmon ladder. The basic idea is that ninja must move up some pegs by "jumping" with a bar. That doesn't sound right. Here, just watch this video.
This looks tough - but who ever said it was easy being a ninja? If it was easy, you would see ninjas all over the place, wouldn't you?
What kind of power would you need to overcome this obstacle? Why is this such a difficult thing to do? Well, a ninja trying to climb this ladder not only has to do something like a pull-up (no easy feat) he has to end the pull-up with enough vertical velocity so that he can be "airborne" long enough for him to move the bar to the next level. Really, this is the part that makes it tough and this is the part that I want to calculate the power for. Let's go.
Video Analysis
I chose the video above because it was the first one I found that had a stationary camera at an appropriate angle. To begin, let me show a plot of the vertical position of this guy's center of mass (which I just estimated). Of course, I am using my favorite video analysis program (and free), Tracker Video Analysis.
Here, the distance units are not meters. I wasn't sure how far apart the ladder notches were, so I just used that as my unit for distance. This means the vertical acceleration during the "in the air" part is about -35.786 ladders per second squared. If I assume this ninja is on Earth, then the vertical acceleration should be -9.8 m/s2. Using this, I can get the distance between jumps at a value of about 0.274 meters. Seems reasonable.
There are two other things I can look at. How long was the ninja in the air and what was his launch speed? From the same graph as above, I can see he was not touching anything for about 0.333 seconds. Also, I can get an estimate for his launch speed by fitting a linear function to the vertical position in the time BEFORE the "jump".
So, about 5 m/s. Ok, a quick check. If you want to launch yourself so that you are in the air for 0.333 seconds. I could also find the velocity by assuming that at the end of this motion, the person would be moving downward with the same speed he started with. Then I could write the following kinematic equation:
With this time interval, I get an initial speed of 1.66 m/s. Ok, not the same as my other method. Actually, let me look at a different time. Let me look at the time the bar is in the air, not the time the center of mass is moving. This looks like this:
This shows the bar is in between notches for about 0.233 seconds. Ok, so an even lower speed is needed. Using the same equation above, this gives a launch speed of just 1.14 m/s. I think I am going to go with this value. Why? Because I feel pretty confident in the time measurement from the video. The other method relies on determining the scale of the video. One small mistake in the scale calculation could through this off.
Energy and Power
Now for some physics (well, more physics). So, you have a ninja. He wants to do two things at the same time. First, he wants to raise his center of mass. Second, during this time he wants to increase his speed. Using the work-energy principle, the work this would require:
Just to be clear, v2 is the speed at the end of the pull. This is also the "launch" speed for the "jump". Also, h is the height that the ninja needs to increase his (or her) center of mass. I already have a value for the speed, and from the video, I will guess a value of h around 0.6 meters (the length from your hands to your shoulders - about). Using these values, I get an estimate for the energy needed to do this of about 508 Joules (assuming a ninja mass of 70 kg). Most of this energy goes into increasing the height of the center of mass (around 400 Joules).
Anyone can do 500 Joules of work. Just imagine climbing a flight of stairs. If a 40 kg child did this at constant speed, this would require about 1000 Joules. See, either this is no big deal or all kids are ninjas. But really, it isn't the work that is a problem. The problem is power. Remember that power is defined as:
So, I just need an estimate for the time it takes to do this pull-up with some extra speed. The longer the time, the lower the power. Let me assume that the ninja accelerates at a constant speed during this motion. In this case, I can use the average velocity (0.83 m/s) over a distance of 0.6 meters would take 0.72 seconds. This seems a bit long. From the video, this motion only takes 0.4 seconds. Ok - well somewhere between these times would give a power requirement of 700 Watts to 1200 Watts. This is just with the arms, so even for a bit this is pretty tough to do. Just for comparison, if this same ninja just did a pull up, it might take around 2 seconds for the motion. This would require 200 Watts.
So, to summarize: 200 Watts = me, 1200 Watts = ninja.
