Vectors

So, let's focus on the mars rover. Opening problem is getting from one point to another.

• Distance it needs to travel? In both 2 and 3D.
  • Add in equation of sphere as a side bonus.
• Give a path of the rover. have them draw it (basically the donkey problem). Then ask them what the two quantities r=vt+b (v and b) have to do with the picture. Ask for the speed of the rover.
• Now define vectors. Define scalar multiplication as well as addition. Define norm or magnitude. Have them practice with these.
• Repeat the first example, with a slightly different example, but this time use the new language. Keep it in 2D for a bit.
• Have them practice with the language more in an abstract setting.
• We now move to 3D. The ideas are the same. First lets practice drawing some objects in 3D (lines, sphere, plane, etc).
• Now have the rover go up a hill in addition to moving. Use the exact same values as a previous problem in 2D, but this time add in the height issue as well.
• Generalize this to give the equation of any object moving in space along a straight line.
• As an exercise, have them use the newly developed formula to give the equation of a line y=mx+b from the past, in the new vector form.

At this point we've solved the from one point to another issue. Other things we should talk about?

• Vector fields? Talk about gravity as a vector field. Have them draw several other vector fields and describe what they think these fields could be used to represent. A sink, as well as a spin field that pushes outwards at the same time, could work. Use software to draw them for sure.
• Angles, dot product, orthogonal
• Projections, work, and kinetic friction. I'd love to teach something like http://hyperphysics.phy-astr.gsu.edu/hbase/frict3.html. If we assume that kinetic friction is proportional to the normal force, then we suddenly have a reason to compute normal forces.
• I see two directions to head, immediately. First, we might as well tackle the problem of pointing the camera at a specific object, while the rover moves. This requires finding the angle between the direction of motion, and the object in question. Note that the location of the rover is constantly changing, so the angle will constantly change too. I'd like a formula for the angle, based on the time elapsed.
• Another instant question is "how much power will we need to get the rover from point A to point B. The answer comes by calculating Work. This is precisely why we need projections. We can (1) compute the work done by gravity, and (2) compute the work done by friction. A great side question here could be having them discover a way to compute the coefficient of friction between a book and an eraser (simple experiment in class). The point at the end would be to compute the total work needed to get from point A to point B. (The solar panels on the rover will provide a specific amount of energy per day, so how much of that energy will be consumed by moving?)
• Vector fields? Should they come up this early? Why not - as a side topic for fun, we might as well. N(t), F(t), F(x,y), f(x,y), f(r(t)) are all things we want them to make sense of and know the difference between them.
• Cross products? Finding area of a parallelogram, and a normal vector to a surface - needed to get the kinetic friction coefficient correctly, though we can't exactly get that complicated at this point, but it wouldn't hurt to show this is a problem we have to solve.
  • What if I gave them the velocity vector (which direction they are headed - or just make them get that), together with a tilt vector (is it flat, or is it tilted from left to right). From there, they have two vectors. I could easily ask them to give me a vector that is orthogonal to both (at which point they will get infinitely many solutions). Have them do this for a specific case first, and then have them do it in general. The solution will be an equation of a line. And in general, they will obtain the cross product formula straight out. Don't mention the name cross product unless someone in class says it. Then have them share the pneumonic trick to remember it. I highly doubt someone will recognize it perfectly.

The question about area hasn't shown up yet. That's fine. We can tackle this later, when it is relevant.

Next Topic is Motion - Parametric Curves

Tracking while you are moving:

• On a moving platform, orbiting mars. You need to point antenna to orbiting satelitte (mars pathfinder?). You need to maintain contact whenever the rover is in sight. This sounds like a need for the satellite dish (paraboloid with focus issue).