Tunnel Problem: Scorpion Gulch and Western Railway is preparing to build a new line through Rolling Mountains. They have hired you to do some calculations for tunnels and bridges needed on the line (Figure 4-6k). You set up a Cartesian coordinate system with its origin at the entrance to a tunnel through Bald Mountain. Your surveying crew finds that the mountain rises 250 m above the level of the track and that the next valley goes down 50 m below the level of the track. The cross section of the mountain and valley is roughly sinusoidal, with a horizontal distance of 700 m from the top of the mountain to the bottom of the valley (Figure 4-6k). a. Write a particular equation expressing the vertical distance y, in meters, from the track to the surface of the mountain or valley as a function of x, in meters, from the tunnel entrance. You can find the constants A, B, and C from the given information. Finding the phase displacement D requires that you substitute the other three constants and the coordinates (0, 0) for (x, y), then solve for D. b. How long will the tunnel be? How long will the bridge be? c. The railway company thinks it might be cheaper to build the line if it is raised by 20 m. The tunnel will be shorter, and the bridge will be longer. Find the new values of x at the beginning and end of the tunnel and at the beginning and end of the bridge. How long will each be under these conditions? 2

Useful Equations for Physics (Part 2) Chapters 14, 20, 21, and 22 C HAPTER 14 Frequency and Period: 1 = F is the frequency, or the number of cycles per second. 1 Hz = 1 cycle per second = 1 s = 1 T is the period, or the time to complete one full second. Kinematic of Simple Harmonic Motion 2 = cos( )