Incorrect Intersection Output of Two Lines

I have to represent a person (Klaus) jumping out of a plane, and then five seconds later a second person (Hans) jumps out of the plane with a parachute to catch the first person (Klaus). I can see on the plot where the two intersect, however, I am unable to output the position and value as an answer. I have tried using the intersect function and setting the two equations equal to eachother but neither have worked.

tklaus = (0:0.001:100)';
thans = (5:0.001:105)';

g = 9.81; % m/s/s
cd = 0.25; %kg/m
mk = 90.7185; %kg
mh = 113.398; %kg sum of mass of hans and parachute (before it is opened)

vk =  @(tklaus) sqrt((g*mk)/cd) * tanh(sqrt((g*cd)/mk)*tklaus);
vh =  @(thans) sqrt((g*mh)/cd) * tanh(sqrt((g*cd)/mh)*thans);


vklaus = vk(0:0.001:100)';
vhans = vh(0:0.001:100)';
xk = zeros(size(vklaus)); %xk is position of klaus in the vertical direction (m)
xk(1) = 4000; %jumps at 4000 m
xh = zeros(size(vhans)); %xh is position of hans in the vertical direction (m)
xh(1) = 4000; %jumps at 4000 m

% Plot to see acceleration plot
% plot(tklaus,vklaus,'b')
% hold on
% plot(thans,vhans,'r')
% hold on
% legend('klaus','hans')
% xlabel('time (seconds)')
% ylabel('velocity (meters/seconds)')

for i = 1:length(vklaus)-1
    xk(i+1) = xk(i) - vklaus(i)*(tklaus(i+1)-tklaus(i));    
end

for i = 1:length(vhans)-1
    xh(i+1) = xh(i) - vhans(i)*(thans(i+1)-thans(i));    
end


%Plot to see velocity plot
 plot(tklaus,xk,'r')
hold on
 plot(thans,xh,'b')
hold on
legend('klaus','hans')
xlabel('time (seconds)')
ylabel('height (meters)')

Answer 1

[Alan Stevens]

After your two for loops put the following

intersectfn = @(t) interp1(tklaus,xk,t) - interp1(thans,xh,t);
t0 = 50;  % Initial guess
tintersect = fzero(intersectfn,t0);
xintersect = interp1(tklaus,xk,tintersect);
disp([tintersect  xintersect])

%Plot to see velocity plot
 plot(tklaus,xk,'r',thans,xh,'b',tintersect,xintersect,'o'),grid
legend('klaus','hans')
xlabel('time (seconds)')
ylabel('height (meters)')

[Brianna]

Thank you for your response, I appreciate the help!

Answer 2

[Jon]

The main difficulty here is comparing Klaus and Hans heights at simultaneous times. Note that the elements of xh and xk are not at simultaneous times. So if we start a stop watch when Klaus jumps, then the time on the stop watch corresponding to the first element of xk is zero, but the first element of xh is 5. So we can't just take the two vectors of heights xh and xk and look element by element to see when they are very close (intersect) as they are not aligned in time.

One way to deal with this would be to rewrite your equations of motion to just use one common time, e.g. a stop watch that starts when Klaus jumps.

Another approach would be to use your data to interpolate values for Hans position at each value of Klaus time, so something like

xhInterp = interp1(thans,xh,tklaus); % interpolate values to Klaus time
plot(tklaus,xk,tklaus,xhinterp)


% define delta as the difference in the two jumpers position (using common time base)
delta = xk -xhInterp;
figure, 
plot(tklaus,delta) % visualize the interception distance
xlabel('time')
ylabel('interception distance [m]')
% find where it the interception distance is closest to zero
% note delta will proably not be exactly zero at your time intervals, but you can approximate
% with closest approach
[deltamin,imin] = min(abs(delta))
tklaus(imin)

[Jon]

It looks like you had already gotten an answer from Alan by the time I posted this. Actually both approaches have similar aspects, Alan's will give a more exact approximation as to the intercept time, but is a little more complicated.

[Brianna]

Thank you for your response, I appreciate the help!