Question

Problem 1. A rifle is fired straight up. Assume there are no cross winds (so this can be treated as a one dimensional problem). Let the vertical direction be z, with its zero at the location of the rifle and increasing as you ascend. Let the acceleration of gravity be g, assumed constant as a function of z. Call the bullet's velocity in the z-direction v. Assume the drag force on the mass-m bullet to be Fa-mav2. (The m is inserted in the definition of Fa for convenience.) la. What is dv/dt equal to? (10 points) 1b. Find the differential equation for v2 as a function of z. That is, dsomething Find that "something." (10 points) 1c. Find the square of the velocity as a function of position, z. Assume v is initially (i.e., at z = 0) equal to some positive velocity, vo. Hint: Solve the differential equation you obtained in part b. This is much easier than solving for the full time dependence. (10 points)

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Answer 1

A rifle is fired straight up Assume there are no cros winds (so this can be treated as aone dimensional problem) F)net = -mg - F_d m dv/dt = -mg - m alpha v^2 dv/dt = -(g + alpha v^2) du/dz = -(g + alpha v^2) 1/2 d/ dz (v^2) + alpha v^2 = -g d/dz (v^2) = -2 (g + alpha v^2) v^2 e^alpha 2 = integral 2g e^alpha 2 dz v^2 e^alpha 2 = 2 g e^alpha 2/alpha + c v^2 = -2g/alpha + e^- alpha 2 C