As part of my continuing obsession over the new Avengers: Age of Ultron trailer, I am going to take a look at the Black Widow. I have no idea what’s actually going on. All I can see is that the Black Widow is riding a motorcycle and scoops up Captain America’s shield.
What is there to analyze? There are two questions. How fast was she going? How hard is it to pick up a shield like that?
It would be simple to just make a plain guess as to the motorcycle’s speed. I mean, she looks like she is driving in a city and in traffic. The would seem to limit her speed to under 60 mph (just as a guess). But you know that baseless guessing isn’t my style. No, I prefer to guess on some wild piece of evidence like the clip from the trailer.
Why not just use my normal kung-fu like video analysis skills? Well, that is my plan. However, this isn’t such a simple clip. The camera angle is very low and the bike is moving mostly towards the camera. This means there is some crazy perspective problems. But isn’t this exactly what I did with the speeder in the Star Wars VII trailer? Yes. Well, no. I think if I just measure the angular size of the wheel, I can use that to get the position. This is a little simpler than the Star Wars speeder analysis.
Here is how this will work. If I know the angular size that an object appears and the actual size of the object, I can find the distance to the object with this equation (more details in my older post on video analysis with angular size).
In this equation, r is the distance from the object to the camera and L is the actual width of the object. θ must be in units of radians and I can find this angular size by assuming the horizontal field of view for the camera is 39.6° (assuming 35 mm film).
The first thing I am going to need is the actual size of something on Black Widow’s motorcycle. Let’s go with the front tire. There is a car that passes by the camera before the motorcycle. I am going to guess it is a 2003 Toyota Corolla (even though this appears to be somewhere in Europe). By looking at a real Corolla, I have estimated that the diameter of the rim is 15.4 inches. I can use that to estimate the diameter of the front wheel.
Here you can see that I took two frames from the scene and overlapped them such that the motorcycle is approximately in the same location as the car wheel. Assuming a constant scale, the motorcycle would have a diameter of 20.1 inches. But wait! This is probably wrong. After a little bit of poking around on the internet, it seems that Black Widow is riding the new Harley-Davidson LiveWire electric motorcycle. Based on a side view of the bike along with a wheelbase of 1468 mm, I get a front tire outer diameter of 0.588 meters. One more note. Why is it so difficult to find the actual size of a tire? Why do tire ratings (120/70-18) have a mixture of measurements in millimeters and inches? That’s messed up.
Ok, one more thing. If I assume the angular field of view of 39.6°, then I can also look at Captain America’s shield. Assuming it is 2.5 feet in diameter (Marvel Wikia) then I can use the angle to calculate the distance to the shield. Putting in values, I get a shield distance of 3.2 meters. That’s a little crazy. It looks like the shield is right there by the camera. Of course the camera could be using a zoom lens and that would make it appear closer than it actually is. If the shield is only 1 meter from the camera then the angular field of view would be just 12°. I will just proceed assuming a range of angular fields of view.
Ok, here is the plot of position vs. time for the motorcycle.
The slope of this line gives the average speed at 36.5 m/s (81.6 mph). That seems a little bit too fast. At least it’s a linear fit.
What about a different estimate (and simpler)? If I assumed the Black Widow was traveling at 36.5 m/s, then during the 0.91 seconds of the clip she would have traveled 33.2 meters (109 feet). That seems pretty far. What if I go the other way? What if I estimate the distance – not by angular size but instead by looking at the strips on the road?
I’m almost positive this scene was filmed in Seoul, South Korea. Looking at images from Seoul, I can estimate the size of white road stripe. Here is a view of stripes from Google Maps.
Screen shot from Google Maps
From this I can see that 9 stripes has a distance of 72.3 meters. This means that from the beginning of one stripe to the start of the next one is 8.03 meters.
Going back to the clip, I need to find out how many white stripes (on the road, not the band The White Stripes) she passes. Based on my best estimate, she passes one complete set of stripes (I assume 8.03 meters) in 0.542 seconds. This would give her a speed of 14.8 m/s (33 mph). That seems like a more realistic speed.
Why is there a large difference in speed for the two methods? It is mostly likely due to a problem with my estimation of the angular field of view for the camera. I’ll be honest. I spent more than an hour looking on Google maps to find that exact location of the shield. If I knew the location I could estimate the angular field of view. Oh well, I’ll try that again at a later date.
Ok, so I’m not too sure about the speed of the motorcycle (but I am going to go with around 15 m/s). Clearly, small set backs like this don’t stop me from calculating things. In this case, I want to estimate the force you would need to pick up the shield. Wouldn’t it just be the same as picking it up off the ground? No. This is different because you are doing two things to the shield. First, you would be picking it up. Second, you would be increasing its speed (up to the speed of the motorcycle).
This is a perfect example of the momentum principle. This says that for a constant net force acting on an object, the following should be true.
Here the Δp represents the change in momentum and the Δt is the time interval that the force acts on the object. Momentum is defined as the product of the object’s mass and its velocity.
Yes, both force and momentum are vectors – but that won’t matter too much in this example. But here you can see the horizontal force needed to pick up the shield. The force will depend on the following:
The change in speed of the shield. I will assume it starts from rest and ends with the same speed as the motorcycle.
The mass of the shield. This is of course not fully known. I previously estimated it at 19.9 kg, but the Marvel Database says that is 12 lbs (5.44 kg).
The time interval of contact. How long did it take the shield to increase in speed?
I have an estimate for the mass as stated above (I will use both values). The speed could be as low as 15 m/s. I can get the time interval from the video. It looks it takes the Black Widow less than 3 frames of video to pick up the shield. This would be around 0.083 seconds.
Before putting in values for stuff, I can write an expression for the force exerted on the shield in one dimension (along the axis of motion).
Using a final velocity of 15 m/s and a shield mass of 19.9 kg, I get a force of 3596 Newtons (808 pounds). That’s pretty high. It might be high enough to dislocate Black Widow’s arm or something. Ok, what if I use the smaller mass for the shield with a value of 5.44 kg? With the lower speed, that would give an average force of 983 Newtons (220) pounds. Maybe she could handle that force for a short time, but it might make her crash on the bike.
Ok, this should be a great life lesson for everyone out there. If you are riding on your motorcycle and you see a shield on the ground, just stop and pick it up. If you try to scoop it up, you might crash. Also, wear a helmet. Sometimes it’s the law, but it’s always a good idea.
