Question

A small rocket having an initial weight of 3000 lb (including 2400 lb of fuel), and initially at rest, is launched vertically upward. The rocket burns fuel at a constant rate of 80 lbs, which provides a constant thrust, T, of 8000 lb. The instantaneous weight of the rocket is w(t) = 3000 - 80t lb. The drag force, D, experienced by the rocket is given by D = 0.005g(dy/dt)^2 lb, where y is distance in ft, and g = 32.2 ft/s^2. Using Newton's law, the equation of motion for the rocket is given by: w/g d^2/dt^2=T-w-D Determine and plot the position, velocity, and acceleration of the rocket (three separate graphs) as a function of time from t = 0, when the rocket starts moving upward from rest, until t = 3. Reduce the second-order ODE to a system of two first order ODEs.

Answer 1

000 fox 2 WE) o0059 9 () 2. ㅂ T- UD dy de? 9 3000.- fola d'y 10 T (coop- Box) - 600-578) S dt consider dy dt 2 dk dt dt 2000 - Go G 9. d3 at - 8000 ~ 3000 + eot - 00059 3 P 어고 5000 + 80k 0-00SgS fi 얘 3000- 8ok 3. 8 dy dh 3 conditione Initial dy y=0, 0 dt

Eular method using Ziti 3 th f (ti,ai) Yiti yit hfa (ti, 3;)

clc clear g = 32.2; 2 3 4 5 6 7 8 9 syms xl x2 % xl corresponds to t, and x2 corresponds to z fi = @(x1,x2) (5000+80*x1-0.005*g*x2+2)/((3000-80*x1)/g); f2 = @(x2) x2; 10 11 12 13 14 15 - h = 0.0001; t = [0:h:3]; N = length(t); y(1) = 0; z(1) = 0; a(1) = fi(y(1),z(1)); for i=2:N z(i) = z(i-1)+h*fi(t(i-1),z(i-1)); y(i) = y(i-1)+h*f2(z(i-1)); a(i) = filt(i),z(i)); end 16 17 - 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 figure; hold on; plot(t,y) xlabel('time') ylabel('position') figure; hold on; plot(t,z) xlabel('time') ylabel('velocity') figure; hold on; plot(t,a) xlabel('time') ylabel('acceleration')

250 200 150 position 100 50 0 0.5 1 2.5 3 N 1.5 time

140 120 100 80 velocity 60 40 20 0.5 1 2.5 3 N 1.5 time

5 55 50 45 acceleration 35 30 25 0.5 1 2 2.5 3 1.5 time