One of my part-time jobs is as an internet investigator. When crazy things happen, people want to know more about that crazy thing. In this case, the crazy thing is a Telsa driving super fast over a railroad crossing. It’s going so fast that the car gets airborne before eventually losing control. Fortunately, it doesn’t seem like anyone was seriously injured, and it is also fortunate that a security camera caught this motion on video. Boom.

Now for some questions.

### How fast was the car traveling?

Normally when I need to find the velocity of an object in a video, I just use my typical video analysis techniques in which I mark the position of the object in each frame. That won’t really work in this case for two reasons. First, the car is much farther away at the beginning of the motion than it is in the middle of the action. Since its distance from the camera changes, its angular size also changes. There is a way to take into account this changing angular size, but it’s not trivial. The second problem is the wide-angle lens of the security camera. Angular sizes for different parts of the video are not true to life. Again, there is a correction for this—but it’s complicated.

So for this jumping Tesla example, I am going to find the velocity with a slightly different method. Instead of marking the position of the car in each frame, I’ll just look at two points in space. If I know the distance between these two points and the time it takes the car to move, I can get the average horizontal velocity (change in position divided by change in time). Fortunately, someone in this particular article describing the event includes the exact location in Google Maps. Now I just need to find recognizable spots in the video that relate to the map.

The distance from the railroad tracks to the signs in the road is about 35.7 meters. Using Tracker Video Analysis, I can get values for the times at these two points with a time interval of 0.801 seconds (assuming the first part of the video plays in “real time”). With these two values, I can calculate the velocity as the following:

Just for comparison, a speed of 44.6 meters per second would be equal to 99.8 mph. Damn! That’s quite speedy and most likely above the posted speed limit—but I am just guessing there.

Quick homework question: Based on the map, there is a section of straight road before the railroad track with a length of about 1 kilometer. What minimum acceleration would this car need to reach the jumping speed in that distance?

### How steep is the road before the track?

Consider the road in front of the track to be like a ramp—which I guess it is. Once the car leaves this part of the road, it is just like a projectile with some initial velocity launched at some angle θ (the inclination of the road). Yes, I am ignoring air resistance.

I don’t know the angle, and I don’t know the launch velocity (I only know the horizontal velocity). If I knew the initial vertical velocity also, I could use these two to find the total velocity and the launch angle. Maybe this diagram will help.

So I just need the initial vertical velocity. The nice thing about projectile motion (without air resistance) is that the vertical motion is independent from the horizontal motion. In the vertical direction, I just have a car that is moving up and then back down with a constant acceleration of 9.8 m/s^{2} (the acceleration of a free-falling object). If the car starts and ends at the same height (an assumption), I get the following equation.

Luckily I already know this time (from the speed analysis above). It’s just a matter of solving for the y-velocity, for which I get 3.9 m/s. Using this with the horizontal velocity, I get a launch angle of about 5 degrees. That seems reasonable. Oh, I can also calculate the total launch velocity now with a value of 44.7 m/s—which is basically the same as before, since it’s launched at such a low angle.

OK, one more homework question. Make a numerical model for the motion of this jumping car that includes air drag. Here is an example of the motion of a soccer ball (with air) that you can use to get started.

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