I don't know how to start this analysis without a spoiler. I can try setting it up with a generic physics question, but if you are behind on the excellent SyFy program The Expanse, you may want to walk away and do something else, like read about why flying at light speed is pretty much impossible unless you're Han Solo.

Still with me? OK. Here's the problem: I've got a spaceship orbiting the sun somewhere in the asteroid belt between Mars and Jupiter, and I want to destroy some other asteroid. Maybe the best way to obliterate an asteroid is to just push it into the Sun. Could I crash this spacecraft into the asteroid to knock it into the Sun?

Yes, this is a pretty tough problem, but I can break it down into three parts: Traveling to the asteroid, colliding with the asteroid, and the resulting trajectory of the asteroid. Before getting started, though, I must make some assumptions. I'll use estimated values from The Expanse, because they've already done the work.

The asteroid is Eros. It follows a circular orbit around the sun (not technically true, but close enough), with an orbital radius of 1.5 AU (where 1 AU, or astronomical unit, is the distance from the sun to Earth). Eros has a mass of about 6.7 x 10 15 kg.

kg. The spacecraft is the *Nauvoo, *a large ship designed for interstellar travel. It's basically a cylinder with a radius of 0.25 km and a length of 2 km. Its starting orbital distance is 2.5 AU.

The Nauvoo has a lot of empty space, so I will estimate its density at 100 kg/m 3 . Using the volume of a cylinder, I get a mass of 4 x 10 10 kg. Wow. Pretty massive for a spaceship.

. Using the volume of a cylinder, I get a mass of 4 x 10 kg. Wow. Pretty massive for a spaceship. I need one final estimate—the thrust of the Nauvoo. With humans aboard, I'd guess you would want an acceleration of 1 g (9.8 m/s2). However, the ship is empty for this mission. Figure it can accelerate at 2 g's.

That's it for the starting assumptions. Now for the physics.

Part 1: Traveling to Eros

I planned to make a numerical model to compute the trajectory and impact speed for Nauvoo. But I won't. It just so happens that orbital mechanics is pretty complicated. You can't simply say, "Point the spaceship toward Eros and fire the engines."

For the best collision with Eros, you want the spaceship to strike head-on. If Eros follows a circular orbit with a radius of 1.5 AU, it would have a speed of about 24,000 m/s. Nauvoo is traveling at around 19,000 m/s. Could the Nauvoo achieve an orbital speed of 24,000 m/s in the opposite orbital direction for the best collision?

With an acceleration of 2g, you'd need only a little more than 30 minutes to go from 19,000 m/s in one direction to 24,000 m/s in the opposite direction. Yes, that seems crazy to me too. But I'll go with it: a head-on collision between Eros and the Nauvoo with each traveling 24,000 m/s.

Part 2: The Collision

I could of course do a simple one-dimensional inelastic collision between the Nauvoo and Eros in which they stick together. Actually, that's a great case for an exam question, but I want to do better than that. Instead I will create something more realistic—a collision that is partly elastic (momentum, but not kinetic energy, is conserved) and it won't quite be in one dimension. I could write this out on paper, but I'll create a numerical calculation because it will look cool.

How do you model a collision? The basic idea is to let the two objects act like springs. When those objects are closer than the sum of their size (so that they overlap), you'll see a spring force pushing them apart. The more they overlap, the greater the spring force. Better yet, I can make this an inelastic collision by using a smaller spring constant when the two objects are moving away from each other. I've gone over the details of such a collision before.

Now for the collision. I have the Nauvoo heading straight toward Eros, but they aren't lined up exactly center-to-center. Here's how the collision will look. Note that Eros is spherical (technically wrong), and the Nauvoo is tiny in comparison. Click "play" to run and the pencil to see and edit the code.

Notice that the program prints out the change in vector velocity for Eros and it's miniscule. The problem is that Eros is something like 10,000 times more massive than the Nauvoo. Although the Nauvoo and Eros will experience the same magnitude for the change in momentum, Eros's mass means a tiny change in velocity. Even if the Nauvoo was traveling at a speed 100 times greater, it still wouldn't do much.

Part 3: Crashing Into the Sun

Since the Nauvoo wouldn't significantly change the velocity of Eros, this part seems silly. But that won't stop me from pondering this. I should note that I've modeled the physics of crashing into the sun before. You might think crashing into the sun would be easy, but no.

Instead of using the change in velocity from my collision calculation, I will assume some super awesome collision gives Eros a change in velocity with a magnitude of 10,000 m/s in some direction. That seems generous. Now I will model two impacts. The first will result a change in velocity directly toward the sun. The second will result in merely slowing Eros down.

This model showing those two impacts, and an undisturbed orbit for comparison. For clarity, the first model is yellow and the second is red.

What happens? You might be surprised to realize that pushing Eros toward the sun actually makes it go farther away from the Sun. Your best option to slow Eros down—but unless you bring it almost to a stop, it simply isn't going to crash into the sun.

I guess that's just as well, because in the end the Nauvoo didn't even collide with Eros. Oops. Spoiler.