A nice application of integration is to compute the distance z() of an object ba

Question

A nice application of integration is to compute the distance z() of an object based on its velocity v(t). Take our bungee jumper as example the velocity is given as v(t) = squareroot gm/c_d tan h(squareroot gc_d/m t) Suppose that we had measurements of velocity at a series of discrete unequally spaced times during free fall. Use the above equation to synthetically generate such information for a 70-kg jumper with a drag coefficient of 0.275 kg/m. Incorporate some random error by rounding the velocities to the nearest integer. Then use cumtrapz to determine the distance fallen and compare the results to the analytical solution z(t) = m/c_d ln[cos h(squareroot gc_d/m t)]. In addition, develop a plot of the analytical and computed distances along with velocity on the same graph.

Explanation / Answer

t=linspace(0,10,10);

% distance computed using integration [i.e., cumtrapz()]
% Then introducing error by rounding off velocity value to nearest integer
v = sqrt(9.807 * 70.0 / 0.275) * tanh( sqrt(9.8 * 0.275 / 70.0) .* t);
v = round(v);
computedDistance = cumtrapz(t,v);

%Analytically computed distance
analyticalDistance = (70.0/0.275) * log( cosh(sqrt(9.8 * 0.275 / 70.0) .* t));

plot(t, computedDistance, 'g', t, analyticalDistance,'b'), legend('Computed Distance', 'Analytical Distance')

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