Question

The upward velocity of a rocket can be computed from the following formula: mo v = u * In( ) – gt mo - qt87 where v = upward velocity, u = the velocity at which fuel is expelled relative to the rocket, mo = the initial mass of the rocket at time t = 0,9 = the fuel consumption rate, and g = the acceleration due to gravity (assumed constant = 9.81 m). If u = 1800m, mo = 160,000 kg, and q = 2600 kg, compute the time at which v = 750m using the bisection method. (Hint: t is somewhere between 10 and 50 s.) Use a stopping criterion of ez = 0.0001.

The MATLAB code should have the outline of this:

m0 = 160e3; % [kg]

u = 1800; % [m/s]

v = 750; % [m/s]

q = 2600; % [kg/s]

g = 9.81; % [m/s^2]

tl = 10; % Lower guess [s]

tu = 50; % Upper guess [s]

es = 0.0001; %Stopping criterion

t_v750 = rocket(m0, g, q, u, v,tl,tu,es)

function t_root = rocket(m0,g,q,u,v,xl,xu,es);

% Inputs:

% m0: initial mass of rocket at time t=0 [kg] (scalar)

% g: gravitational constant [m/s^2] (scalar)

% q: fuel consumption rate [kg/s] (scalar)

% u: velocity at which fuel is expelled relative to rocket [m/s] (scalar)

% xl: Initial guess for the lower root (scalar)

% xu: Initial guess for the upper root (scalar)

%

% Outputs:

% t_root: time when the rocket reaches the specified velocity [s] (scalar)

%

%% Write your code here:

Answer 1

m0 = 160e3; % [kg]

u = 1800; % [m/s]

v = 750; % [m/s]

q = 2600; % [kg/s]

g = 9.81; % [m/s^2]

tl = 10; % Lower guess [s]

tu = 50; % Upper guess [s]

es = 0.0001; %Stopping criterion

t_v750 = rocket(m0, g, q, u, v,tl,tu,es)
function t_root = rocket(m0,g,q,u,v,xl,xu,es)

% Inputs:

% m0: initial mass of rocket at time t=0 [kg] (scalar)

% g: gravitational constant [m/s^2] (scalar)

% q: fuel consumption rate [kg/s] (scalar)

% u: velocity at which fuel is expelled relative to rocket [m/s] (scalar)

% xl: Initial guess for the lower root (scalar)

% xu: Initial guess for the upper root (scalar)

%

% Outputs:

% t_root: time when the rocket reaches the specified velocity [s] (scalar)

%

%% Write your code here:
f = @(x) u*log(m0/(m0-q*x))-g*x-v;
a=xl;
b=xu;
iter = 0;
  
if f(a)*f(b)>=0

disp('No Root')

else

prev = (a+b)/2;
p=a;
while (abs(f(p))>es)
prev=p;

iter =iter+ 1;

p = (a+b)/2;

if f(p) == 0
p
q=1
break;

end

if f(a)*f(p)<0

b = p;

else

a = p;

end
if(iter==100)
disp('the required accuracy is not reached in 50 iterations');
end
end

end
t_root=p;
end

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