Download Introduction to Computational Physics I - Homework 4 | PHYS 105 and more Assignments Physics in PDF only on Docsity! PHYS 105: Introduction to Computational Physics Spring 2009 Homework #4 (Due: May 14, 2009) 1. Reconsider the 1-dimensional motion of a particle moving under a harmonic force: a(x) = −k x , again with k = 4 and x = 0, v = 1 at t = 0. However, now friction also acts on the particle, producing an acceleration proportional to, and always opposite, the velocity: af = −αv . For small values of α, the mathematical solution to the equations of motion may be shown to be of the form x = Ae−bt sin ω′t . (a) For α = 0.1, and taking time steps δt = 0.001, determine the period T and hence ω′ by finding the first time (after t = 0) the particle crosses x = 0 with v > 0 (using linear interpolation, as usual, to refine the answer). (b) By considering the decrease in amplitude from one maximum (or minimum) to the next, determine the ratio r by which the amplitude decreases from one peak to the next and hence the value of b for this α. 2. Do in-class exercise 6.2. Turn in your program (parts 1 and 2), three plots (parts 3–5), and clearly state the value of the slope you obtain in part 5. 3. (a) A particle moves in two dimensions under the combined effects of gravity and air resistance. The components of its acceleration are ax = −αvx , ay = −g − α vy , where g = 9.8m/s2. The particle is launched at time t = 0 from x = 0, y = 0 with a speed of v0 = 100 m/s at an angle θ to the horizontal. It is desired to have the projectile hit a horizontal target running from x = 300 to x = 320 m at a height of y = 200 m, from above — i.e. with vy < 0. (Note: To determine whether this occurs, first interpolate the trajectory to the value of x when y = 200 m, then check whether x lies in the desired range.) (a) In the case of no air resistance (α = 0), determine, to one decimal place, the minimum and maximum values of θ in degrees that will accomplish the goal. You may do this numerically or analytically. If you do it numerically, find the angle to within 0.1◦.
