Hints on Project 2

Problem 1 (Roller Coaster)

1. The equations for x and y are simply parametric equations, not equations in polar coordinates. If the use of theta is confusing to you, just replace it by t. (I did not use t in the problem because theta does not correspond to time).
2. You should think of the shape parameter sigma as a fixed (but unknown constant). Thus, when you calculate the curvature, your answer will be an expression that involves sigma. Your task will then be to choose sigma to keep the roller coaster safe.
3. There are many different ways you can calculate the curvature. See the table on page 811 of your textbook. There is one formula in particular that makes the computation very easy. Other methods are valid, but may be considerably more work.
4. To find the normal component of acceleration, use the formula a_N=K(speed)^2. Many of you have been confused by the formula in the book that says a_N=K(ds/dt)^2. In that formula, recall that s(t) represented the arclength function (as a function of time t), so that (ds/dt) was actually the speed.
5. Somebody tried to calculate speed from the formulas for x(theta), and y(theta). However, this is impossible to do because theta does NOT represent time. (So just use the speed given to you explicitly in the problem).
6. When you are done, your answers for sigma should be between 0 and h/2. If you get something different, then you've done something wrong. (Most common cause is a mistake in conversion of miles/hr into feet/sec.)

Problem 2 (level surfaces)

1. To find the maximum elevation, you should be able to find this by looking at your graph. You don't need to set the gradient equal to zero.
2. To answer part 5, the question can be rephrased "is there a path you can take from the starting point to the ending point such that the highest point and the lowest point on the path differ by no more than the stated limit."