Clearly, I am not a professional blogger. I am an amateur. This is because I was under the impression that only amateur bloggers could compete in the blogging olympics. When did they change these rules? Anyway, Adam Weiner did a physics-based analysis of the latest Star Trek movie trailer. Here is the trailer:
In the trailer (oh, spoiler alert) a young Kirk jumps out of a car before it goes over a cliff. It does look odd, and that is why I had intended to analyze it. In Adam's analysis, at PopSci.com the basic approach was:
- Take the initial velocity of the car (from the clip)
- Assume the car is slowing due to friction over a distance of 30 meters
- Calculate the speed of the car as Kirk jumps out
- Calculate the acceleration of Kirk in order to stop in 5 meters
- Use this acceleration (and the mass of Kirk) to calculate the force he needs to grab the edge of the cliff.
- The result is that he finds Kirk has to have superhuman strength. (which of course, he does)
When I saw this video, I thought of taking a slightly different assumption. From the video, it looks like as Kirk leaves the car he is not moving too fast with respect to the ground. He could do this with super-human jumping ability.
On further inspection of the video, it seems like there is enough in the clips to do a video analysis of the motion with Tracker Video - my favorite free video analysis tool.
To begin my video analysis, I will look at the scene that shows the car from the top slowing down near the cliff. I am not a car buff, but it seems like a 65ish corvette (from wikipedia). I can use this to scale the video. The length of the car is 175.1" according to idavette.net or 4.4 meters. Here is a shot of the scene I am talking about followed by postion-time data from Tracker video:
First, notice that in one dimension I also fit a quadratic function to the data, but it is not accelerating that much during this scene. I can fit a linear function to get a good idea of the speed. The car is not a point object, so it was also rotating. I chose a point near the middle of the car. It has motion in both the x- and y-direction. Using the two dimensions of velocity, I can get the total velocity. The two components of velocity are just about 4.4 m/s.
So, from this clip it seems the car is not traveling oh-too fast. Another thing I can get from that video clip is the distance of the car from the edge of the cliff. At the last frame, the car is 5.7 meters from the cliff. I am not sure if Adam used video analysis for his blog post, but he stated the distance Kirk started from the cliff as about 5 meters. If he did not use the video, this was a very good estimation.
Now, looking at the second scene - where Kirk is jumping out of the car. There is a problem with this clip in that it is clearly in slow motion. However, I can use a trick. I know what Kirk's vertical acceleration should be and I can use this to find the time between frames.
So, using Video Tracker, I obtained data for Kirk's head in each frame. Really, I should have used his center of mass since he is rotating, but I just want a rough estimate. I am assuming the scene has a minimal zoom changing and I used the back of the corvette to scale the video. Here is the data for Kirk's head:
Although the graph says t is in seconds, it's not. Not the best data, but anyway, Kirk's head should have an acceleration of -9.8 m/s2. This would give a coefficient of -4.9 in front of the t2 term. If I call the time scale in the clip ts, then the following would be true (note the term in front of t2 in the fit is -.159):
Now I can look at the velocity of the car and Kirk in the second shot. Note the time here is still in movie time.
So the x-velocity of the two objects are:
Are these speeds reasonable? Well, I had the car in the first scene going 13 mph and now it is 4.5 mph. It could have slowed down in between the two shots, which is possible. I measure that Kirk is 3.3 meters from the edge of the cliff. Anyway, I will just go with 9 mph for the speed of the car. If that is the case, then Kirk had to jump at about 10 mph. Can someone jump like this? Well, If someone did jump at 10 mph (about 4.5 m/s) how high would they jump? Here I can use the work energy relationship with the person and the Earth as the system. In this case, there is no work done, but there is a change in gravitational potential energy.
When a person jumps, his (or her) initial speed is something (in this case 4.5 m/s) and his (or her) final velocity is 0 m/s (at the highest point). I can use this this to find how high a person would go jumping this high.
I can't jump 1 meter high (really, I can't) but I am sure there are some people that can. So, maybe all in all this shot isn't so bad. If Kirk is only moving like 1 mph after the jump out of the car, he won't need superhuman strength to stop him. I must say, I thought this looked completely whacked out when I first saw it also.
This is not to diminish the awesomeness of Kirk.