AVENGERS
OK, THERE’S A new trailer for Avengers: Age of Ultron. A new trailer means a new analysis. I am going to pick the scene that shows the Avengers jumping out of a jeep during some type of fight in a snow covered forest. Why are they in the forest? Who are they fighting? Why are some of them riding in a jeep? I have no idea about these questions. All I can do is analyze the motion of these superheroes. That’s what I’m going to do.
Let me start by looking at the motion of Black Widow during this scene. The camera does pan a bit, so I will use Tracker Video Analysis to compensate for this background motion. Here is the same scene with a stationary camera.
The one other thing I need is a scale. Thor and Black Widow are fairly close to each other so I will use the size of Thor (Chris Hemsworth is 1.9 meters tall). With that, I get the following for the horizontal motion of Black Widow.
It’s clear that the end of the scene is in slow motion. I bet they used the iPhone 6 to record this part of the movie. If I assume the first part of the motion is in real-time, I can fit a linear function to the position-time graph and get the horizontal velocity with a value of 8.88 m/s (about 20 mph).
When a person is off the ground, there is essentially only one force on that person – the gravitational force. Since this force is in the vertical direction, there is zero force in the horizontal direction. And this is why the horizontal velocity is constant. Now, going back to the data I can look at the slope of the later part of the function. This says the horizontal speed is 1.41 m/s. However, this isn’t in “seconds” but some slowed down time unit (I will call it s‘). Since I know the horizontal speed should be constant, I can set the before and after speeds equal to each other.
This means the slow motion part of the video is 6.3 times slower than real-speed (assuming the beginning is in real speed).
Now let’s look at the vertical motion. It isn’t as pretty for three reasons. First, it’s accelerating in the vertical direction – so no straight lines. Second, there is that time in frame rate in the slow motion part. Third, the Black Widow doesn’t move up and down that much. This means that any smaller error in marking a point looks more dramatic.
There is one way I can fix the problem with the changing time – don’t plot time. I could just plot the trajectory of Black Widow (y vs x instead of y vs t). Since the trajectory doesn’t show the time, you don’t have problems with the slow motion vs. regular motion. However, dealing with a trajectory equation is a little bit more difficult that y vs. time. Let me fix it another way.
Since I already calculate the change in frame rate from the horizontal motion (6.3 times slower for the slow motion part), I can use that to fix the time values in the later part of jump. Now I should have a vertical motion plot with consistent time values. Here’s what that looks like.
Now I can fit a quadratic equation to this data (for the part after they leave the jeep) and find the vertical acceleration. By comparing this function to the kinematic equation I can see that the value in front of t2 should be 1/2 the acceleration. This means that the vertical acceleration in this scene is -4.74 m/s2. Yes, you do remember your introductory physics class. You remember that this value on Earth is supposed to be -9.8 m/s2. The acceleration is half what it should be. Why? Here are some options.
The scale is wrong. Maybe Black Widow is actually twice the size that I guessed. This would make the apparent acceleration too small. Of course that would mean she would be 10 feet tall. Perhaps this is evidence that Ant Man is in the movie and the Avengers use Pym Particles to increase their size.
Maybe they aren’t jumping. Maybe they are flying. Perhaps they all have the ability to somewhat fly which decreases their vertical acceleration.
What if my assumption about the first half of scene being in real time was wrong? What if the whole jump is in slow motion?
Ok, I am going to fix the time problem. Let’s say that the acceleration is off by a factor of 2 because the video is not in real speed. That means I can write the two accelerations equal to each other (but with different time units) just like before.
This means the first part of the scene is 1.41 times slower than real speed and the second part is 8.89 times slower. I can use this to fix the initial horizontal speed with a value of 6.3 m/s (14 mph). But what about the vertical velocity? This is a little bit tougher to estimate. Looking at the data from the vertical motion, the Black Widow only increased her vertical position by about 20 cm. She barely jumped at all. If you want to jump up a height of 0.2 meters, you would need an initial vertical velocity of about 2 m/s.
But in the end, what does all of this say about The Avengers: The Age of Ultron? Not much – just that I’m ready for the movie to be out. Oh, there will be more analysis of this trailer coming soon. Wait – do you need extra homework for this video? What if you look at the other heroes. How fast are they moving? You will probably need to rescale the video for each person since they are different distances from the camera.
