*See National Math Standards for this Challenge.*

**
Q: How fast will the impactor spacecraft be moving when it hits Tempel 1?**

In one way this is a bit of a trick question. Since both the comet and the impactor spacecraft will be moving in nearly the
same orbit at the time of impact, what is important really is how fast the impactor spacecraft will be traveling relative to the
comet. As an example of the importance of this difference, consider two cars involved in a rear-end collision. If both cars are
moving at the time they collide, it isn't the velocity of the first car or the second car that would be used to calculate the energy
of impact, it's the difference between them. So a car traveling at 60 mph striking a car traveling 55 mph transfers the same
amount of energy at collision as a car traveling 5 mph striking a stationary car.

Another complication that has to be taken into account is the fact that the comet and impactor spacecraft will be moving
through three-dimensional space, instead of the "one dimensional" space of the car example. Fortunately this can
be fixed by thinking of the motion one piece at a time, using vectors.

If you assign the same x-y-z set of axes for both objects, their movement can be broken down into three components, an
x-component, a y-component and a z-component. This gives you the velocity that the object is moving along that axis. The
overall velocity of the object can be determined by simply adding up the vectors. Since the three vectors will be at right
angles to each other (since they're along the three right-angle axes), we can just use the three dimensional form of the
Pythagorean theorem:

Looking back at the car example, let's say that the cars were moving along the x-axis. This means that their velocities
(in mph) along the three axes would be:

**Car 1:** (60, 0, 0)

**Car 2:** (55, 0, 0)

This means that the overall velocities of the cars would be:

**Car 1:** = 60 mph

**Car 2:** = 55 mph

Okay, enough hints. The last piece of information you need is the x, y, and z velocities (in kilometers per second) for the
comet (27.09, -11.40, -5.46) and the impactor spacecraft (18.41, -12.67, -0.16). From these, can you determine the velocity
of the impactor spacecraft relative to the comet at the time of impact?

**
A: You need to do a bit of manipulation to answer this question correctly.**

Looking at the Deep Impact mission, the x, y, and z velocities (in kilometers per second) for the comet Tempel 1 at the time of
impact are (27.09, -11.40, -5.46). This means the overall velocity of the comet is:

And the x, y, and z velocities (in kilometers per second) for the impactor spacecraft at the time of impact are
(18.41, -12.67, -0.16). This means the overall velocity of the impactor spacecraft is:

At this point you may think; "Hey, wait a minute! Your mission description says the impactor spacecraft will be moving at 10.2
km/s. You messed up somewhere!" Think back to the car example from earlier. The *relative* velocity of the slower car to
the other was only 5 mph, even though it was moving at 55 mph itself.

Now, you might be tempted at this point to say that the comet is just 7.5 km/s (29.9-22.4) faster than the impactor spacecraft,
so the relative velocity of the impactor spacecraft will be 7.5 km/s. Unfortunately, this does not take into account the fact that the
two objects will not be heading in exactly the same direction, like the cars were in the earlier example.

So, instead, to determine the *relative* velocity of the impactor spacecraft, you have to subtract each component
(x, y, and z) of the comet from the impactor spacecraft, and then run the same type of 3D Pythagorean theorem calculation on the
resulting relative components.

This works even in the "one dimensional" car example. The relative velocities of the second car will be: (55-60, 0-0, 0-0) or
(-5, 0, 0). This means the overall relative velocity of the second car to the first is:

= 5 mph

Again looking at Deep Impact, the relative velocities of the impactor spacecraft then will be:
(18.41-27.09, -12.67-(-11.40), -0.16-(-5.46)) or (-8.68, -1.27, 5.30). This means the overall relative velocity of the impactor
spacecraft is:

= 10.2 km/s

So, the impactor spacecraft will transfer the same amount of energy to the comet as it would if the comet were stationary and
the impactor spacecraft were moving at 10.2 km/s.

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