Physics 263: Problem Set #2

1. (BTM 1.6.6) To find P(i), generalize the formula you used in BTM 1.6.5 (from PS#1) from two terms to n terms. Check the hint for that problem for an example of how to calculate the average of a quantity in terms of the probabilities. The problem is saying that P(i) is of the form (constant)*e^-beta*E(i), with the same constant for all i. How would you determine that constant? The last part, which asks you to use BTM 1.6.5 is asking you to do two things: show that m = d(ln Z)/d(beta h) and then to evaluate this explicitly and show that you get mu*tanh(beta mu h) again.
2. (BTM 1.6.8) Recall that you need to check the sign of the second derivative to determine if a "stationary point" (that is, a place where the first derivative is zero) is a maximum, a minimum, or an inflection point. Feel free to check your algebra and sketch with MATLAB.
3. (BTM 1.6.10) The key to this problem is drawing a good picture before you write down any equation, with quantities labeled (e.g., the angles in the problem, and all of the distances, whether you know them or not). Note that the problem is to minimize the time for the life guard to get to the swimmer. So if you can construct a function that describes how the total time depends on one of the angles, it is this function you need to minimize. Consider how long it takes on land and how long in the water (your picture should include a sample path, with angles theta1 and theta2). You should be able to find two equations involving the two unknowns theta1 and theta2 and other constants (v1, v2, L, d1, and d2).
4. (BTM 1.6.12) We did problem 1.6.7 in class on Friday two ways. The first way was with a substitution and the second one was by applying implicit differentiation. You are to show here that these two ways give the same answer for dy/dx if y and x are related by the equation for a circle.