Following shows how to write this in MATLAB.
syms theta theta_1 V_A V_Bx;
W_A=100;
m_A=W_A/32.2;
W_B=20;
m_B=W_B/32.2;
L=15;
theta_0= deg2rad(70);
% Solve the system of three equations
one = W_A*L*(cos(theta)-cos(theta_0))-1/2*m_A*V_A^2-1/2*m_B*V_Bx^2+W_A*L;
two = m_B*V_Bx+m_A*(V_Bx+L*theta_1*cos(theta));
three = -V_A^2+(V_Bx+ L*theta_1*cos(theta))^2+(L*theta_1*sin(theta))^2;
[theta_1,V_A,V_Bx] = solve(one,two,three);
% Plot the velocity of the trolley and the speed of the rider as a function of the angle θ of the first half of a full swing
fplot(V_Bx, [-70,70], 'Linewidth',2)
hold on
fplot(V_A, [-70,70], '-.*c')
hold off
% Plot the velocity of the trolley and the speed of the rider as a function of the angle θ of the second half of a full swing
fplot(V_Bx, [-70,70], 'Linewidth',2)
hold on
fplot(V_A, [-70,70], '-.*c')
hold off
The above can take a long time to plot because there are 12 solutions given by solve() functions. To speed up the computation, you can convert the functions to numeric function handle using matlabFunction() and then use fplot().
