This section deals with changing coordinates, and also deals with objects that don't move a straight line.

The rover was always moving in a straight line. Maybe we don't want that. Maybe we want it to curve. To be be honest, the simplest way to program the rover would be to move in piecewise straight lines, then change direction with a pivot. Give them an example of a piecewise smooth curve and have them draw it. Then have them state the velocity at each point along the path.

The equations we have for the rover all depend on several things. Where should the start point be? Which direction is right, left, up, down, back, forward, etc. What does 1 unit represent (ft, m, mile, km, etc). How does changing this change our equations? Is there a simple way to transfer from one system to another? Introduce linear changes of coordinates. To illustrate, work with circles and parabolas. For these examples, just focus on u=ax+b and v=cy+d. Don't rotate in the examples, unless the students ask in class. Toss the whole section on hyperbolas (they can graph them later in life).

Not all things move in a straight path. To get the rover to mars required moving along a highly nonlinear path. Let's turn our attention to motion along paths that are not straight.

  • Give them a circular path. Give it two ways. Have them draw it in the xy plane. then have them draw it in the xyt space. Use Mathematica to construct the plots as well to check yourself.
  • Swap to an ellipse. have them guess what the object is before they draw (with a reason for their guess), and then plot the points. Have them develop a Cartesian equation for it as well.
  • Do this for a line in 3D (review)
  • Now do this for a parabola.
  • Now turn the tables and ask them for equations, provided details of what they should have. Turn the tables much earlier on.
  • What about a cycloid? This might be a great exercise for in class. The hurricane example is perfect.

Don't get too derailed above, because we want to get to derivatives.

  • Using the examples above, ask about velocity, speed, and acceleration.
  • Develop tangent lines (why - motivation would be nice).
  • Normal acceleration might be nice (if we want to do curvature as well)?????
  • Arc length is perfect.
  • Talk instantly about mass, center of mass, centroid, average height, etc.
  • I do really like the total charge, given the charge density (great for physics students - and fits perfectly after a discussion about mass).
  • Here is a great place to follow up with questions about work. This is precisely why we care about the tangent line. So maybe we don't need a tangent line, rather we just need the derivative for Force dot displacement. Add these up over the length of the path you go, and you get the work done. Perfect.
  • This is a great place to leave a question for later, namely as you move along the surface how do you find the total force from friction? We would need to know precisely the tilt in addition to the direction of motion, or really we just need a normal vector to the surface. We'll come back to this later.

Next topic - Polar Coordinates

Side note: Do we need hyperbolas at all? Probably not, unless we want to talk about quadric surfaces. We should probably introduce them, but I'd rather just let them plot the stuff with software. Unless we end up with a use..... I do want to talk about ellipses, but that's easy to model with a change of coordinates.