Differential equation problem Please answer fully and show work so it is easy to

Question

Differential equation problem

Please answer fully and show work so it is easy to understand. Will rate, thanks!

[1] A squash weighing 1 lb is shot vertically upward from the earths surface with an initial velocity of 1000 ft/sec. The air resistance in pounds is numerically equal to 0.0001v^2=(.01v)^2 where v is the velocity of the squash in ft/sec.

a) Choose your positive axis in the direction of the initial velocity and determine the velocity of the pumpkin as a function of time.

b) Determine how long it will take for the pumpkin to reach it's highest point.

Explanation / Answer

a) Let initial velocity be v0.

   Air resistance is Fair = - ßv2 where ß= 10-4

   F = -mg - ßv2 , m = mass of pumpkin, g = 9.8m/s2

But F = ma => ma = -mg - ßv2 and a = dv/dt

mdv/dt = -mg - ßv2 or -m/ mg + ßv2 dv = dt

dt =( -m/ mg + ßv2) dv

Integrating, t = m/gß ( tan -1 v0/mg/ß - tan-1 v/mg/ß )

Inverting, v(t) = mg/ß tan( tan -1 ßvo2/mg - (gß/m) t)

                  = 313 tan( tan -1 3.194 - .0313t) = 313 tan (2pi/5 - .0313t)

b) Let vt = mg/ß

   We need v > 0 at any point in time for the ball to come down.

   This implies, t <= vt/g tan-1 (vo/vt)

   tmax = 313/9.8 tan-1(1000/313) = 31.94* 2pi/5 = 31.94* 1.2673 = 40.48 seconds

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