Predicting distribution of random variable Y=AX+B from the knowledge about distribution of random variable X: a case study of projectile motion

close all

clear

figure

g=9.81; %gravity

V=5; %m/s projection speed

theta=pi/3; %projection angle

x(1)=0;%initial x

y(1)=0;%initial y

k=2;

for t=0.001:0.001:1

x(k)=V*cos(theta)*t+x(1);

y(k)=-0.5*g*t^2+V*sin(theta)*t+y(1);

k=k+1;

end

plot(x,y,'.')

xlim([0 3])

ylim([-1 2])

hold on

quiver(x(1),y(1),V*cos(theta),V*sin(theta),'AutoScaleFactor',0.2)

xlabel('x')

ylabel('y')

plot(0,0,'o','MarkerSize',20,'MarkerFaceColor','r')

The projectile is projected with velocity of V at an angle from horizon. The projectile should be projected from origion but the mechanism that places the projectile at the origion is not accurate. Its error in x direction follows a Gaussian distribution with mean and standard deviation of (sigma_x) and its error in y direction follows a Gaussian distribution with mean and standard deviation of (sigma_x). These errors are independent from one another. The following simulation code allows you to investigate the statistical distribution of the location of the projectile after 1 second from projection.

• x(i,1)=sigma_x*randn;%initial x
• y(i,1)=sigma_y*randn;%initial y

for i=1:M

x(i,1)=sigma_x*randn;%initial x

y(i,1)=sigma_y*randn;%initial y

k=2;

for t=0.001:0.001:1

x(i,k)=V*cos(theta)*t+x(i,1);

y(i,k)=-0.5*g*t^2+V*sin(theta)*t+y(i,1);

k=k+1;

end

subplot(1,3,1)

plot(x(:,1),y(:,1),'o')

title('Initial conditions')

xlim([-4*sigma_x 4*sigma_x])

ylim([-4*sigma_y 4*sigma_y])

axis square

xlabel('x(0)')

ylabel('y(0)')

subplot(1,3,2)

title('Projectile trajectory at each trial')

plot(x(i,:),y(i,:),'.')

hold on

end

xlim([0 3])

ylim([-1 2])

axis square

xlabel('x')

ylabel('y')

subplot(1,3,3)

plot(x(:,end),y(:,end),'bo')

title('Final position after 1 sec')

hold on

axis square

xlim([2 3])

ylim([-0.9 -0.2])

figure

mu = [V*cos(theta) -0.5*g+V*sin(theta)]; % found from analyzing equations of motion

Sigma = diag([sigma_x^2,sigma_y^2]); % found from analyzing equations of motion

rng('default') % For reproducibility

R = mvnrnd(mu,Sigma,M);

plot(R(:,1),R(:,2),'r+') %these are points that we predict from our analysis

hold on

axis square

plot(x(:,end),y(:,end),'bo') %these are points that the simulaiton of 500 trial cases give us

axis square

xlim([2 3])

ylim([-0.9 -0.2])

xlabel('x(1)')

ylabel('y(1)')

figure

clear x y

M=500;%number of trials

g=9.81; %gravity

V_command=5; %m/s projection speed

theta=pi/3; %projection angle

sigma_v=0.2;

e=sigma_v*randn(1000,1);

for i=1:M

x(i,1)=0;%initial x

y(i,1)=0;%initial y

V=V_command+e(i);

k=2;

for t=0.001:0.001:1

x(i,k)=V*cos(theta)*t+x(i,1);

y(i,k)=-0.5*g*t^2+V*sin(theta)*t+y(i,1);

k=k+1;

end

subplot(1,2,1)

plot(x(i,:),y(i,:),'.')

title('Projectile trajectory at each trial')

hold on

end

xlim([0 3])

ylim([-1 2])

axis square

hold on

xlabel('x')

ylabel('y')

subplot(1,2,2)

plot(x(:,end),y(:,end),'o','MarkerSize',8)

axis square

xlim([2 3])

ylim([-1.1 -0.1])

hold on

figure

mu = [V_command*cos(theta) -0.5*9.81+V_command*sin(theta)];

Sigma = [cos(theta) sin(theta)]'*[cos(theta) sin(theta)]*sigma_v^2;

rng('default') % For reproducibility

R = mvnrnd(mu,Sigma,M);

plot(R(:,1),R(:,2),'r+')

hold on

plot(x(:,end),y(:,end),'o','MarkerSize',8)

axis square

xlim([2 3])

ylim([-1.1 -0.1])