As shown in Figure 5.3.11, a plane flying horizontally at a constant speed v₀ drops a relief supply pack to a person on the ground. Assume the origin is the point where the supply pack is released and that the positive x-axis points forward and that positive y-axis points downward. Under the assumption that the horizontal and vertical components of the air resistance are proportional to (dx/dt)² and (dy/dt)², respectively, and if the position of the supply pack is given by r(t) = x(t)i + y(t)j, then its velocity is v(t) = (dx/dt)i + (dy/dt)j. Equating components in the vector form of Newton’s second law of motion,

gives

(a) Solve both of the foregoing initial-value problems by means of the substitutions u = dx/dt, w = dy/dt, and separation of variables. See the Remarks at the end of Section 3.2.

(b) Suppose the plane files at an altitude of 1000 ft and that its constant speed is 300 mi/h. Assume that the constant of proportionality for air resistance is k = 0.0053 and that the supply pack weighs 256 lb. Use a root-finding application of a CAS or a graphic calculator to determine the horizontal distance the pack travels, measured from its point of release to the point where it hits the ground.

Figure 5.3.11

Transcribed Image Text:

dx\2 i+ dv m dt mg - k dt j