Who Hits the Water First? Investigating Free Fall Bungee Jump Physics

[Saman Salike]

Two girls are standing on a bridge over a river. One girl falls freely and the other ties a rope to her legs. Both jump at the same time from the bridge. The length of the rope is more than the vertical distance between the bridge and the river so that the rope never becomes taut till she touches the river. Who goes into the river first ?

 

[Ken G]

Neither. If the rope is never taut, it supplies no force.

 

[jbriggs444]

Two girls are standing on a bridge over a river. One girl falls freely and the other ties a rope to her legs. Both jump at the same time from the bridge. The length of the rope is more than the vertical distance between the bridge and the river so that the rope never becomes taut till she touches the river. Who goes into the river first ?

Depends on the rope.

If we assume a massless rope then Ken G's response is correct. If we assume a massive rope coiled on the bridge then a retarding force will apply and the tethered girl will hit the water last. If we assume a massive but supple rope coiled in the girls hands and allowed to unwind freely then both hit together.

 

[Ken G]

The problem said the rope is not taut, so cannot apply a force even if it has mass. The rope is in free fall also, apply the equivalence principle.

 

[jbriggs444]

No, the problem said the rope is not taut, so cannot apply a force even if it has mass.

Incorrect.
If the rope is massive then the transition from coiled-on-the-bridge to falling-with-the-girl requires a non-zero impulse.

 

[Ken G]

Incorrect.
If the rope is massive then the transition from coiled-on-the-bridge to falling-with-the-girl requires a non-zero impulse.

Yet the problem clearly stated the rope is not taut, so the equivalence principle is clear on this. Imagine a person in zero gravity, attached to a non-taut rope. Now accelerate something the rope is attached to. No effect on the person until the rope goes taut. If you change the problem to make the rope be taut, of course you will change the answer.

 

[jbriggs444]

No. The equivalence principle is clear on this. Imagine a person in zero gravity, attached to a non-taut rope. Now accelerate something the rope is attached to. No effect on the person until the rope goes taut.

I do not agree. However, the disagreement may center on the meaning of the term "taut". Let us return to the bridge situation.

There is a rope with one end attached to the bridge and the remainder coiled and resting near the attachment point. We unwind a short length of rope and tie it to the girl. At this point we agree, I think, that the rope is not "taut". Instead, it is "slack".

The girl jumps off the bridge horizontally. Some amount of rope is dragged with her. We can ignore the horizontal forces involved in this transition because they are horizontal and do not affect the vertical motion we care about. At this point the rope is still not "taut". However it is also not entirely "slack". Loops of rope are being pulled from the coil horizontally. There is non-zero tension in the rope. A small tension, but a tension nonetheless.

The girl falls. The rope is still unreeling from the coil and the unreeled portion is still under tension. It is presumably feeding over the edge of the bridge so that the tension felt at the girl is largely vertical. This has an effect on the girl, slowing her fall.

 

[Ken G]

Remember that if the rope has mass, it does not support a constant tension along its full length. The tension will be zero by the time you get to the girl, until the rope goes taut. Some sort of signal must propagate along the rope, changing its shape as the connectedness at one end is communicated to the other end. When that signal arrives, there can be a force at the end, but not before. That signal is responsible for the tension, but also requires tension in order to propagate. It does sound like a tricky problem that strains the simple meaning of "taut," but we can imagine that "not taut" means the signal hasn't arrived yet.

 

[jbriggs444]

If you force the rope to unreel, then it must at some point go taut, changing the problem. The problem states the rope is never taut, so can never unravel. If you make the length of the rope that is slack smaller than the distance to the river, the rope will always go taut, and the problem states otherwise. Why pretend this problem is harder than it is?

As I said, your understanding of the term "taut" is different from mine. I understand "taut" from the original problem statement as emphasizing that the rope is long enough to reach the water. The fact that it is tied to the bridge is, thus, irrelevant. But if we are contemplating a massive rope, that is not the only detail that matters. Why make the problem easier than it is? Picky details are (sometimes) what make toy problems fun.

But let us take a different scenario. The rope dangles, supported at the tie-off point to the bridge on the one end and at the tie-off point at the girls feet on the other. The girl jumps and we ask whether the dangling rope pulls her into the water more rapidly than gravity alone. This time the answer is easier. The section of rope dangling from the girl is in free fall, just like the girl and is under zero tension. The girls hit the water together. The somewhat interesting bit of this scenario is that the section of rope dangling from the bridge is motionless and is under tension. At the bottom of the sag in the rope is a point where the rope is undergoing an instantaneous acceleration. This acceleration allows for a transition between the zone under no tension and the zone under tension.

Edit: Note that the chain-drop video contradicts the above assertion. There is non-trivial physics involved at the bottom of the sag. In at least some circumstances, tension can propagate through that point. [If I remember correctly, one explanation involves angular momentum and leverage as each link crosses the boundary].

 

[Ken G]

As I said, your understanding of the term "taut" is different from mine. I understand "taut" from the original problem statement as emphasizing that the rope is long enough to reach the water. The fact that it is tied to the bridge is, thus, irrelevant. But if we are contemplating a massive rope, that is not the only detail that matters. Why make the problem easier than it is? Picky details are (sometimes) what make toy problems fun.

I agree, but the force of gravity applies to the rope as well. So the only way its motion could be altered is if one end of the rope is attached to the bridge. But that only alters the motion of the rope to a point where a signal can propagate along the slack rope, and the arrival of that signal is the simplest interpretation of the meaning of "taut." Nevertheless, I agree that the propagation of that signal is itself an interesting and difficult problem-- just not the one being asked about.

But let us take a different scenario. The rope dangles, supported at the tie-off point to the bridge on the one end and at the tie-off point at the girls feet on the other. The girl jumps and we ask whether the dangling rope pulls her into the water more rapidly than gravity alone. This time the answer is easier. The section of rope dangling from the girl is in free fall, just like the girl and is under zero tension. The girls hit the water together. The somewhat interesting bit of this scenario is that the section of rope dangling from the bridge is motionless and is under tension. At the bottom of the sag in the rope is a point where the rope is undergoing an instantaneous acceleration. This acceleration allows for a transition between the zone under no tension and the zone under tension.

I agree that the tension in a sagging rope is an interesting issue, related to the shape that a sagging rope adopts. But when the girl jumps, the tension at her end of the rope becomes zero, and remains zero until a signal that straightens the rope arrives at her. The rope at her end does not immediately know if it is attached to the bridge at the other end, and when the signal that it is attached arrives, we can say that signal brings with it the concept of "tautness." But it is true that the rope does not need to be straight when that signal arrives, because the rope has mass. The shape of a falling rope that is attached at one end is not an easy problem I'm sure.

 

[jbriggs444]

The shape of a falling rope that is attached at one end is not an easy problem I'm sure.

I had not considered the situation with a sagging rope with non-negligible horizontal extent. That is an interesting problem. But I think we are in agreement about the physics of the situation, if not about the exact intent of the problem that was posed.

 

[Ken G]

Even the rope with no horizontal extent is interesting, in the calculation of the time dependence of the place where the rope turns upward. I think that would be a classic example of what I mean by propagation of "tautness" along the rope-- the propagation along the rope of the turnup point. But if that turnup point never reaches the girl, her motion will never be altered, and we can say the rope is "not taut."

 

[russ_watters]

Yet the problem clearly stated the rope is not taut...

The way it is worded implies to me a vagary, where I can't tell if it referred to the entire rope or left open the possibility of the rope being pulled off the bridge. So I like the idea of considering both of those possibilities.

 

[beamie564]

The way it is worded implies to me a vagary, where I can't tell if it referred to the entire rope or left open the possibility of the rope being pulled off the bridge. So I like the idea of considering both of those possibilities.

How about the possibility of the girl tied to the rope (who's mass is significant) falling first? Like in this video

 

[Ken G]

Yes, the issue is all about what is meant by a rope or chain being "taut". To me, it just means it is supporting tension at the place it attaches to the girl, so if there were slack present, it would straighten as the tension force arrives at the end and the rope goes taut. A signal arrives along the rope when the tension arrives. We can see that happening in the falling chain in the video-- at first there is no difference in the motion because the tension in the chain at the mass is zero, which we could notice by putting a slack wiggle into the chain as we dropped it, and waited for when that wiggle straightened (when the chain went "taut"). We should be able to see the tautness signal propagate along the rope, and arrive, as tension, at the mass. The tension originally in the chain, before the mass is released, should quickly go to zero as the two masses fall together, then later the one mass gets a jerk as the signal arrives and tension reappears at the end of the chain. But I can see that "never goes taut" could also mean that the chain is never hanging down completely straight, and the OP could be interpreted like that, in which case the rope could support tension prior to being "taut", like in the video. It may be the intention of the question that the girl with the rope hits the water first, as in the video. Or if the rope is being pulled off the bridge, that girl arrives second. Or if the rope is always entirely slack in the sense that it is dropped with the girl and the signal that it is attached at the other end never arrives at the girl, then they hit at the same time. So it is indeed interesting to consider all these possibilities.