Download Classical Mechanics with Maxima by Todd Keene Timberlake, J. Wilson Mixon PDF

By Todd Keene Timberlake, J. Wilson Mixon

This e-book publications undergraduate scholars within the use of Maxima―a machine algebra system―in fixing difficulties in classical mechanics. It capabilities good as a complement to a regular classical mechanics textbook. in terms of difficulties which are too tough to resolve through hand, laptop algebra structures which could practice symbolic mathematical manipulations are a priceless device. Maxima is especially appealing in that it's open-source, multiple-platform software program that scholars can obtain and set up at no cost. classes discovered and features built utilizing Maxima are simply transferred to different, proprietary software program.

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Example text

For k D 5 we get a different optimal angle. (%i) find_root(dh(theta*deg,5),theta,0,90); (%o) 22:208 A larger value of k results in a smaller optimal launch angle. How do we interpret this physically? Recall that k D y0 =hmax . So a larger value of k means the projectile starts from a launch point that is higher, relative to how far up the projectile would travel if fired straight up. When the projectile is launched from a higher point it makes sense that a smaller (more horizontal) launch angle will maximize the range.

The best way to ensure that you are using a sufficiently small step size is to run the calculation with smaller and smaller step sizes until you don’t see any change in the solution. The output from rk is a list in which each element contains a value for the independent variable as well as the corresponding values for the functions for which we solved. In the code below we apply rk to our example of a water drop fired from a water gun, but this time with quadratic resistance. Recall that the launch angle is 45ı and the initial speed is 15 m/s.

63. 6 m/s. In our example above the rain drop reaches an appreciable fraction of its terminal speed before hitting the ground. The only reason it doesn’t make it all the way to terminal speed is that it hits the ground before it can do so. If we were to let the drop fall for a longer period of time we would find that it reaches its terminal speed (or very close to it). To verify this fact we can evaluate the rain drop’s velocity at t D 100 s. (%i) v(100); (%o) 128:56 The speed of the rain drop after 100 s of fall is very nearly equal to the drop’s terminal speed.

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