calculate the flight path of a projectile with and without air resistance. For s
ID: 3679124 • Letter: C
Question
calculate the flight path of a projectile with and without air resistance. For simplification, we will assume that the projectile is small and spherical in shape. Your task is to write a MATLAB programs that calculate the flight path, range and flight time for several values of coefficients of resistance of the projectile, and plot the results.
coeficient of resistance= 0-.08 with step size of .001
initial velocity=600
release angle =60
If we assume that the projectile is launched from the surface of the Earth, i.e., at y0 = 0, then the altitude of the projectile as a function of time is
Y(t)=(-1/2)*g*t^2 +Vt;
where g = 9.81 m/s2 is the gravitational constant, t is time (in seconds), and V = v0 sin (in meters per second). The distance of the projectile from the launching position as a function of time is
x(t)=U*t
where U = v0 cos . The velocity of the projectile in the horizontal direction, i.e., parallel to the x-axis, is
u(t)=U
and the velocity in the vertical direction, i.e., parallel to the y-axis, is
v(t)=-gt+V
If we include the effects of air resistance, then the altitude of the projectile as a function of time can be approximated as
y(t)=-gt/k+((kV+g)/k^2)*(1-e^(-kt))
where k is the coefficient of resistance (with the unit 1/s). The distance of the projectile relative to the initial position as a function of time is
x(t)=(u/k)(1-e^(-kt))
The velocity of the projectile in the horizontal direction is
u(t)=ue^(-kt)
and in the vertical direction
v(t)=Ve^(kt)+(g/k)*(e^(-kt)-1)
create
1. A MATLAB function, flightpath.m, capable of calculating the flight path of a projectile, with and without air resistance, given an initial speed v0, an angle of departure relative to the horizontal, and a coefficient of resistance k.
2. A script, main_flightpaths.m, which uses the function flightpath.m to calculate and the function plot_flightpaths.m to plot the following: a. Altitude, y, as a function of distance, x (see Fig. 2) b. Altitude, y, as a function of time, t c. Horizontal velocity, u, as a function of time, t d. Vertical velocity, v, as a function of time, t For these plots, assume v0 = 600 m/s, = 60 deg., and k = [0 0.005 0.01 0.02 0.04 0.08] s-1 .
3. A script, main_range.m, which uses the function flightpath.m to calculate and function plot_range.m to plot the following: a. Range as a function of k b. Total flight time as a function of k For these plots, assume v0 = 600 m/s, = 60 deg., and k from 0 to 0.08 s-1 with step size of 0.001 s-1 .
Explanation / Answer
Please find below the MATLAB code which can be used to calculate flight path, range and flight time:
flightpath.m:-
clear all;
global rkcoef ad76
global tdata aoadata bankdata
global req mu omega mass sref
% define angular conversion factors
rtd = 180.0 / pi;
dtr = pi / 180.0;
% radius of the earth (kilometers)
req = 6378.14;
% gravitational constant of the earth (km**3/second**2)
mu = 398600.4415;
% earth rotation rate (radians/second)
omega = 7.2921151467d-5;
% initialize rkf78 function
rkcoef = 1;
% ---------------------------------
% define propulsion characteristics
% ---------------------------------
% aerodynamic reference area (km**2)
sref = 2.499091776e-4;
% read atmospheric density data
[fid, ad76] = read76;
% ------------------------------
% read flight controls data file
% ------------------------------
m = csvread('sts_cr.csv');
tdata = m(:, 1);
aoadata = dtr * m(:, 2);
bankdata = dtr * m(:, 3);
ndata = size(tdata);
% --------------------------------------
% define initial flight path coordinates
% --------------------------------------
% altitude (kilometers)
xalt = 121.92;
% relative velocity (kilometers/second)
vrel = 7.80288;
% geographic declination (degrees)
dec = 0.0d0;
% geographic longitude (degrees)
elon = 0.0d0;
% azimuth (degrees)
azim = 90.0d0;
% flight path angle (degrees)
fpa = -2.39780555815067d0;
% vehicle mass (kilograms)
mass = 92079.25;
% -------------------------------
% load initial integration vector
% -------------------------------
% geocentric altitude (kilometers)
yi(1) = xalt;
% longitude (radians)
yi(2) = dtr * elon;
% geocentric declination
yi(3) = dtr * dec;
% relative speed (kilometers/second)
yi(4) = vrel;
% flight path angle (radians)
yi(5) = dtr * fpa;
% flight azimuth (radians)
yi(6) = dtr * azim;
fprintf(' program demo_fpeqm ');
fprintf(' initial flight path coordinates');
fprintf(' -------------------------------');
fprintf(' altitude %14.4f meters', 1000.0 * yi(1));
fprintf(' velocity %14.4f meters/second', 1000.0 * yi(4));
fprintf(' declination %14.4f degrees', rtd * yi(3));
fprintf(' longitude %14.4f degrees', rtd * yi(2));
fprintf(' azimuth %14.4f degrees', rtd * yi(6));
fprintf(' flight path angle %14.4f degrees ', rtd * yi(5));
% data generation step size (seconds)
deltat = 5.0d0;
% final simulation time (seconds)
tfinal = tdata(end);
% number of differential equations
neq = 6;
% truncation error tolerance
tetol = 1.0d-10;
tf = 0.0d0;
nplot = 0;
% integrate equations of motion and create data arrays
while (1)
h = 1.0d0;
ti = tf;
tf = ti + deltat;
if (tf > tfinal)
tf = tfinal;
end
% evaluate lift and drag coefficients
alt = yi(1);
alpha = interp1(tdata, aoadata, ti, 'cubic');
bank = interp1(tdata, bankdata, ti, 'cubic');
yf = rkf78 ('fpeqms', neq, ti, tf, h, tetol, yi);
% load data arrays for plotting
nplot = nplot + 1;
% simulation time (seconds)
xplot(nplot) = ti;
% altitude (meters)
yplot1(nplot) = 1000.0 * alt;
% angle-of-attack (degrees)
yplot2(nplot) = rtd * alpha;
% bank angle (degrees)
yplot3(nplot) = rtd * bank;
% longitude (degrees)
yplot4(nplot) = rtd * yf(2);
% declination (degrees)
yplot5(nplot) = rtd * yf(3);
% velocity (meters/second)
yplot6(nplot) = 1000.0 * yf(4);
% flight path angle (degrees)
yplot7(nplot) = rtd * yf(5);
% azimuth (degrees)
yplot8(nplot) = rtd * yf(6);
if (tf == tfinal)
break
end
% reload integration vector
yi = yf;
end
fprintf(' final flight path coordinates');
fprintf(' -----------------------------');
fprintf(' altitude %14.4f meters', 1000.0 * yf(1));
fprintf(' velocity %14.4f meters/second', 1000.0 * yf(4));
fprintf(' declination %14.4f degrees', rtd * yf(3));
fprintf(' longitude %14.4f degrees', rtd * yf(2));
fprintf(' azimuth %14.4f degrees', rtd * yf(6));
fprintf(' flight path angle %14.4f degrees ', rtd * yf(5));
% plot altitude
plot(xplot, yplot1);
title('STS Maximum Crossrange', 'FontSize', 16);
xlabel('simulation time (seconds)', 'FontSize', 12);
ylabel('altitude (meters)', 'FontSize', 12);
grid;
print -depsc -tiff -r300 demo_fpeqm1.eps;
% plot speed
figure;
plot(xplot, yplot6);
title('STS Maximum Crossrange', 'FontSize', 16);
xlabel('simulation time (seconds)', 'FontSize', 12);
ylabel('speed (meters/second)', 'FontSize', 12);
grid;
print -depsc -tiff -r300 demo_fpeqm2.eps;
% plot flight path angle
figure;
plot(xplot, yplot7);
title('STS Maximum Crossrange', 'FontSize', 16);
xlabel('simulation time (seconds)', 'FontSize', 12);
ylabel('flight path angle (degrees)', 'FontSize', 12);
grid;
print -depsc -tiff -r300 demo_fpeqm3.eps;
% plot azimuth angle
figure;
plot(xplot, yplot8);
title('STS Maximum Crossrange', 'FontSize', 16);
xlabel('simulation time (seconds)', 'FontSize', 12);
ylabel('azimuth angle (degrees)', 'FontSize', 12);
grid;
print -depsc -tiff -r300 demo_fpeqm4.eps;
% plot declination
figure;
plot(xplot, yplot5);
title('STS Maximum Crossrange', 'FontSize', 16);
xlabel('simulation time (seconds)', 'FontSize', 12);
ylabel('declination (degrees)', 'FontSize', 12);
grid;
print -depsc -tiff -r300 demo_fpeqm5.eps;
% plot longitude
figure;
plot(xplot, yplot4);
title('STS Maximum Crossrange', 'FontSize', 16);
xlabel('simulation time (seconds)', 'FontSize', 12);
ylabel('longitude (degrees)', 'FontSize', 12);
grid;
print -depsc -tiff -r300 demo_fpeqm6.eps;
% plot declination versus longitude
figure;
plot(yplot4, yplot5);
title('STS Maximum Crossrange', 'FontSize', 16);
xlabel('longitude (degrees)', 'FontSize', 12);
ylabel('declination (degrees)', 'FontSize', 12);
grid;
print -depsc -tiff -r300 demo_fpeqm7.eps;
% plot angle-of-attack
figure;
plot(xplot, yplot2);
title('STS Maximum Crossrange', 'FontSize', 16);
xlabel('simulation time (seconds)', 'FontSize', 12);
ylabel('angle-of-attack (degrees)', 'FontSize', 12);
grid;
print -depsc -tiff -r300 demo_fpeqm8.eps;
% plot bank angle
figure;
plot(xplot, yplot3);
title('STS Maximum Crossrange', 'FontSize', 16);
xlabel('simulation time (seconds)', 'FontSize', 12);
ylabel('bank angle (degrees)', 'FontSize', 12);
grid;
print -depsc -tiff -r300 demo_fpeqm9.eps;
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