There are a couple of things to notice. First, after the collision DART is moving backwards, because it bounced. Since velocity is a vector, that means that it will have a negative momentum in this one-dimensional example.
Second, the kinetic energy equation deals with the square of the velocity. This means that even though DART has a negative velocity, it still has positive kinetic energy.
We just have two equations and two variables, so these equations aren’t impossible to solve—but they’re also not trivial. Here’s what you would get if you did the math. (If you really want all the details, I have you covered.)
Illustration: Rhett Allain
Using the values for DART and Dimorphos, this gives a final velocity of 1.46 mm/s. That’s twice the recoil velocity for the inelastic collision. Since the DART spacecraft bounces back, it has a much larger change in momentum (going from positive to negative). This means that Dimorphos will also have a larger change in momentum and a larger change in velocity. It’s still a tiny change—but twice something tiny is bigger than tiny.
Elastic and inelastic collisions are just the two extreme ends of the collision spectrum. Most fall somewhere in between, in that the objects don’t stick together but kinetic energy is not conserved. But you can see from the calculations above that the best way to change the trajectory of an asteroid is with an elastic collision.
Looking at images of Dimorphos after the collision, it seems that there is at least some material ejected from the asteroid. Since the debris moves in the opposite direction of DART’s original motion, it appears that the spacecraft partially bounced back, showing the increase in the change in Dimorphos’ momentum. That’s what you want to see if your goal is to budge a space rock. Without any ejected material, you would have something closer to an inelastic collision with a lower asteroid recoil velocity.
How Can We Measure the Result of the Impact?
As you can see from the previous example, the best-case scenario would change the velocity of the asteroid by just 1.34 millimeters per second. Measuring a velocity change this small is quite a challenge. But Dimorphos has a bonus feature—it’s part of a double asteroid system. Remember, it’s orbiting its bigger partner, Didymos. That’s one of the reasons NASA chose this target. The key to finding the effect of a spacecraft crashing into Dimorphos will be measuring its orbital period, or the time it takes for the object to make a complete orbit, and seeing if it has changed following the collision.
Dimorphos orbits Didymos according to the same physics that make the moon orbit the Earth. Since there is a gravitational interaction between them, Didymos pulls Dimorphos toward their common center of mass—a point much closer to the center of Didymos, because it’s larger. This gravitational force would cause the two objects to eventually collide if they both started from rest. But that’s not the case. Instead, Dimorphos has a velocity that’s mostly perpendicular to this gravitational force, which causes it to move in an orbit around the center of mass. It’s possible (but not absolutely necessary) that this orbit is circular.
