There are a few things to note. First, after the collision, the DART is moving backwards, because it bounces. Since velocity is a vector, that means it will have negative momentum in this one-dimensional example.
Second, the equation of kinetic energy corresponds to the square of the velocity. This means that even though the DART has a negative velocity, it still has positive kinetic energy.
We only have two equations and two variables, so these equations aren’t unsolvable — but they’re not trivial either. This is what you will get if you do the math. (If you really want to know all the details, I protected you.)
Illustrated by: Rhett Allin
Using the values for DART and Dimorphos, this gives a final speed of 1.46 mm/s. That’s twice the recoil velocity for inelastic collisions. Since the DART spacecraft turned back on, it has a much momentum changes larger (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 small change – but something twice as small is bigger than tiny.
Elastic and inelastic collisions are just two poles of the collision spectrum. Most fall somewhere in between, where the objects don’t stick together but kinetic energy is not conserved. But you can see from the above calculations that the best way to change an asteroid’s orbit is by elastic collisions.
Looking at the images of the Dimorphos after the collision, it seems that at least some matter was ejected from the asteroid. Since the debris moves in the opposite direction of the DART’s original motion, it appears that the spacecraft is partially bouncing back, indicating an increase in Dimorphos’ momentum change. That’s what you want to see if your goal is to move a space rock. Without any material being ejected, you would have something closer to an inelastic collision with a lower asteroid recoil velocity.
How can we measure the outcome of the impact?
As you can see from the previous example, the best case would change the asteroid’s velocity by only 1.34 millimeters per second. Measuring this small change in velocity is quite a challenge. But Dimorphos has an extra feature — it’s part of a binary asteroid system. Remember, it revolves around its larger counterpart, Didymos. That’s one of the reasons NASA chose this target. The key to figuring out the effect of a spacecraft hitting Dimorphos would be to measure its orbital period, or the time it takes the object to make a complete orbit, and see if it changes after the collision. .
Dimorphos orbits Didymos following the same physics that causes the moon to orbit the Earth. Because of their gravitational interaction, Didymos pulls the Dimorphos toward their shared center of mass — a point much closer to Didymos’ center, because it is larger. This gravitational force will cause the two bodies to eventually collide if they both start at rest. But that’s not it. Instead, Dimorphos has a velocity mostly perpendicular to this gravitational force, causing it to orbit around its center of mass. It is possible (but not absolutely necessary) that this orbital is circular.
