Solving for Unstretched Length in Jump Problem: ##l_0 = 97.5m##

[STEMucator]

Homework Statement

The $m = 75kg$ man jumps off the bridge with $v_1 = 1.5 m/s$. Determine the unstretched length $l_0$ of the cord in order that he stops momentarily above the surface of the water. The stiffness is $k = 80 N/m$.

Picture: http://gyazo.com/80220984e620ed5fc0c58c8207e79ba9

Homework Equations

The Attempt at a Solution

This is one of those problems they don't give the answer for and I'm wondering if this is okay.

I drew my FBD stick figure with the Datum plane located through position A. I then used conservation:

$T_1 + V_1 = T_2 + V_2$

Since I stuck the datum plane at the top where the man is, he currently posses no potential energy at all, only kinetic.

In the final phase, the man has lost all of his kinetic energy and it is now in the form of potential. So we have:

$\frac{1}{2} (75) (1.5)^2 + 0 = 0 - (75)(9.81)(150) + \frac{1}{2} (80) (150 - l_0)^2$

Which is quadratic in $l_0$. Solving I obtain $l_{0_1} = 203m$, which makes no physical sense at all since it exceeds the jumping height. $l_{0_2} = 97.5m$, which seems reasonable.

Does this look okay?

 

[verty]

I believe your answers are correct. He falls for $l_0$ meters at which point the rope becomes taut. Over the next $150 - l_0$ meters, two forces act on him, tension and gravity. Each force does work but $W_{tension} - W_{gravity} = E_k$. This gives a pretty complicated formula that can be solved for $l_0$. With some computer help I got the same two answers: 202.55m and 97.45m.

 

[willem2]

The reason you get the extra root, is because (150-l_0)^2 is positive if l₀>150, so the potential energy is positive. This is true for a spring with length l₀ , but not for a bungee cord which has 0 potential energy if the cord is longer than the height of the fall.

Your computation is rather dangerous, because it will produce the right result if the bungee jumper comes to a stop at the bottom, but if the bungee jumper does not come to a stop, you might still get a result.