The Asteroid in The Good Dinosaur Travels at Half the Speed of Light

Stop right there. Yes, I know this is just a movie. And yes, I know it's a kids' movie. But that won't stop me from looking at this asteroid in The Good Dinosaur. I think the basic idea of this movie is to consider what would happen if the dinosaurs were not destroyed by an asteroid. So, of course the trailer shows this asteroid moving past the Earth---but not hitting it.

How about a quick analysis?

Fortunately, the trailer has a nice shot of the asteroid (I guess it's an asteroid) moving past the Earth. Since I know the size of the Earth, I can use this to set the distance scale in the video and then use Tracker Video Analysis to get a plot of the position of the asteroid.

Here is the position vs. time plot for that object.

That looks fairly linear. This means that the slope of the linear function would give the speed of the asteroid. Looking at the fitting function, I get an asteroid speed of 1.66 x 10⁸ m/s (371 million mph). That's fast, but is it too fast? Let me drop another number: 2.998 x 10⁸ m/s. This is the speed of light. That means the asteroid is traveling at 55.3 percent the speed of light (or as we would write it 0.553c where c is the speed of light).

Just for fun, we can also look at the motion of the asteroid as seen from the ground as it is shown in the video. I don't know the scale, so the distance is measured in pixels.

I am pleasantly surprised that this is not a linear function. As an object moves past the Earth, it's distance from the viewers changes. This means that it should have a higher apparent speed when closer to the Earth. I suppose that's what is happening here---but I will leave a detailed analysis up to you.

You might think that super fast things are just like normal things---but super fast. That's not true. It turns out that our usual models for moving objects don't work when those objects are moving close to the speed of light. In particular, we need to consider the energy. For low speed objects (like a bullet or a turtle---both are low compared to the speed of light), we can write the kinetic energy as:

And then we could add the rest mass energy (mc²) to get the total energy. But when objects are moving closer to the speed of light, we can't just write the kinetic energy as a separate term. Instead, we have to write the kinetic energy as the stuff after the mass energy.

I have an estimate for the speed of the asteroid, but what about the energy? Let's say that this is the same object that could have caused the dinosaur extinction---the Chicxulub impactor. But how massive was this object? It seems that there are several estimates, but I am going to go with an asteroid size of 10 km (spherical). Using an asteroid density of 3.0 g/cm³, I get a mass of 1.57 x 10¹⁵ m/s.

Using this mass and the speed from the video, I can calculate the kinetic energy of the asteroid. I get a value of 2.8 x 10³¹ Joules. This is significantly higher than the estimated impact energy of the Chicxulub impact at about 1.0 x 10²⁴ Joules (yes, that is 1 million times more energy). If the Chixculub was energetic enough to cause mass extinctions, what would a million of these asteroids do?

You might be thinking that I couldn't possibly put this much thought into an analysis of a simple trailer. Oh, I could do even more. However, I'm going to save these other calculations as a homework assignment. Here are your questions.

• Doppler Effect. When an object is moving towards an observer, that observer will see the object as producing a shorter wavelength (blue shifted). When moving away, the object appears at a longer wavelength (red shifted). For an asteroid speed of 0.5c, what should the color look like as it moves past the Earth?
• Relativity. When an object is moving near the speed of light, weird things happen. As the object gets closer to the viewer, you would detect light from that object (see it) sooner than if it was farther away. What should an asteroid actually look like moving that fast? Really, I have no idea about the answer to this question.
• Realistic Speed. Suppose the asteroid started in the very outer part of the solar system and then accelerated towards the Earth due to the gravitational pull of the Sun. How fast could this asteroid be moving if it started from rest? I am guessing that this velocity value will be significantly lower than what I have measured.
• Fix the frame rate. Find a reasonable value for the asteroid speed. For this case, how long would it take to pass the Earth? See if you can fix the video. Should there be a deflection due to the gravitational interaction with the Earth?
• The view from Earth. How about an analysis of the asteroid as seen from Earth (in the clip). What can we learn from this? Does the motion in that scene agree with the motion of the asteroid as seen far from the Earth?
• Why is it glowing? Should the asteroid be glowing like that? Why?
• Energy to destroy the Earth. Using my estimate for the kinetic energy of the asteroid, could this completely destroy the Earth? How much energy would it take to gravitationally separate all the mass of the Earth?

OK, that's your homework. I would just like to point out that I could have answered most (but not all) of these questions in a single blog post---that would have been overkill.