By Gary Turner
For newbie to complex
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We might now derive the Euler-Lagrange equation for this problem, but we observe its translation-invariance in the variable 0 [1, p. 262], and we obtain a first integral with no intermediate steps: Here E has the dimensions of energy, whence E/F has the dimensions of length. Direct substitution into (3) gives a first-order equation for the unknown curve: Moreover we expect a "turning point" on this curve where we achieve minimal or maximal distance from the origin. Thus we let (r0, 0o) be the coordinates of this point, and we note that dr/dd = 0 at (r0, 00).
We first consider paths with x'(z) ^ 0 (' means d/dz) with the car on the inside of the track. The forces on the track due to gravity and curvature are given by respectively, with r, = the radius of curvature. By setting the total force [=fK(z) +/(z)] equal to a constant multiple of the car's weight, mgc, we find that Given x'(z) = tan 0(z) [0(z) = the angle the track makes with the vertical], (2) becomes Equation (3) is separable. Its solution is 9 MECHANICS Given an initial slope Jt'(z 0 ) = tan 00, then A = Vz0(c - sin 00).
It is well known that a harmonic function does not have maxima or minima. Hence 0 is unstable. Safe Car Following Problem 87-14*, by VINCENT SALMON (Menlo Park, California). It has been recommended by some highway safety officials that for reasonable safety in a lane of traffic with auto AI following A2, Ai should remain at least fo seconds behind A 2 . Thus starting at any given time, AI should take a time interval of at least /0 seconds to reach the location that A2 occupied at that time. Now suppose that the speeds and spacing are initially constant and optimal, and that A2 accelerates.