Question about momentum and larger masses

[khkwang]

Question about momentum and larger masses (ice tubing, and why we have to let go)

Okay, so my friends and I went snow tubing recently where the track was as such:

_
..\
...\<---hill
...\........./
...\........../<---stopping end hill
...\___________________________/

Where the hill at the end is to slow you down and stop you.

The people there instructed us that if we were going as a group, to let go of each other's tubes, before reaching the speed bump because if we stayed together as a group, our speed would be much greater, and will throw us over the end.

This is where I'm confused. It seems to me that there would be no difference whether or not you stayed as a group. My reasoning being:

Assume there are 3 people, each with a mass of 1kg
After going down the hill, the 3 people as a single mass, M, has a velocity of 2m/s

Then p=M$$v_{1}$$=6kgm/s

After splitting apart, momentum is conserved,
so p=(m$$v_{2}$$) + (m$$v_{2}$$) + (m$$v_{2}$$)
Which results in each person having the same velocity as before, 2m/s.

Now, assuming a frictionless surface, the only thing slowing down the tubers would be the hill at the end: the force of gravity. Which would slow down the tuber at a rate of g*cos$$\vartheta$$, regardless of the mass.

Therefore, regardless of whether the 3 tubers stayed as one group or not would not make a difference in how far up the hill they go at the end.

Is my argument sound?

Also, in the practical setting, I did observe that the ones who stayed together as a group really did go further up the hill than the ones who split apart though.

The reason for this that I can come up with is that friction is somehow the culprit. but no matter what, my friend keeps believing that the reason has something to do with momentum and how the single tubers only have a momentum of 2kgm/s, while the group has 6kgm/s, and so the group is more difficult to slow down.

Is he correct or am I?

In either case, could someone explain why?

 

[cepheid]

I think that you're right. Although an object with three times greater momentum is harder to slow down, this is exactly compensated for by the fact that the force involved is gravity, which increases in proportion to the mass. Hence, the mass dependence cancels and the acceleration is the same for any object going up the incline, as you have noted. I can't really explain what you observed.

 

[Matterwave]

Friction should not mess with the analysis you put together either, since it's proportional to the normal force which is proportional to the mass as well.

The only force that would change would be air resistance. People who hang together have less air resistance since the people in front block the air from hitting people in back. I'm not sure if this effect would actually give rise to your observations though, since I think this effect is probably quite small.

 

[diazona]

I'm sort of grasping at straws here, but when you have a group of snow tubers hanging on to each other, the distance between the front ones and the center of mass of the group is larger for larger groups. And the physical prediction that the height doesn't change based on the size of the group only applies to the center of mass. I suspect that if you were very careful to look at the center of mass of each group, rather than the people on the front, you'd see that the heights they reach are not so dissimilar.

To put it another way, if you're all holding on to each other, the tubers at the front of the group are going to "steal" some momentum from the tubers at the back of the group, so they wind up going higher than they would if they were isolated. The stolen momentum gets transferred through the couplings between the tubers (i.e. people's arms). And if one person falls over the edge, well, all bets are off and they might drag the whole group along

 

[khkwang]

Matterwave, I came to the same conclusion today about the friction too.

I seems like the only other force acting on tubers would be air resistance, but I never knew air resistance would be that strong.

diazona, I'm not sure this explains what I saw. When sticking together, the tubers' final speed was much faster and more likely to hit the net at the very back than compared to when the tubers separated, who definitely slowed down at a greater rate.

Also, my friend still won't believe me and said my diagram was inaccurate. Which it is, but only because drawing it was a little complicated, so here goes again.\
.\
..\
...\............../
...\..........____...hay.../
...\........../...\...V.../
...\ _____________________________/...\_______/

so the section labeled "hay" is ice mixed with hay to provide more friction.

Am I correct in assuming that even in this situation, neither the friction from the ice or the friction from the hay is to blame for the difference in final speeds?

 

[Laura681]

Hey khkwang, its so weird that you’re thinking about this exact question. I’m actually the reckie for Snowshoe Mountain in Virginia (we’re techies and reckies. techies operate the magic carpet, reckies knock the kids down). We tell the groups to let each other go when they get to the bottom so they don’t go up the stop hill and into a metal wire fence. But I never understood why we tell them that, it didn’t make any sense to me, and yet every time they let go at the end they’d be fine. At the same time, all the groups that held on to each other crashed horribly into the fence. I’ve had some time to think this over, and this is what we have to work with:

(larger group)

-greater force of gravity

-greater force of friction, (I don’t know where you tubed, but at Snowshow, on the flattened part of the ride before the end hill, we sketch our ice. That means we engrave lines in the ice, perpendicul to the direction of the hill, so riders have something to slow them down at. They usually have something like that everywhere by tubing spots)

-greater amount of momentum

-greater air resistance, (I think matterwave said there’d be less air resistance in a larger group, which is only true if you’re talking about the SUM of all resistances. That is to say, in total, a large group of people keeping together would have a lower TOTAL amount of air resistance compared to a dispersed group. Think about shotgun pellets and a canon ball. The spray of a shotgun would have a greater total air resistance than a cannon ball (we’re assuming to total mass of the spray equates the mass of the cannon ball), but any ONE particular pellet in the spray will have LESS air resistance compared to the cannon ball. So I figure, its not air resistance that slows each rider down when individual riders get to the end hill)

-equal velocity (compared to individual, because groups break up after max speed is reached)

(smaller group)
-smaller force of gravity
-less momentum
-less friction
-less air resistance (individually)
-equal velocity

As we can see, everything works against the larger group. If anything, the larger group should slide peacefully to a stop, and the groups that divvy up should keep flying. The larger group has more gravity, more friction, and more air resistance. However, the groups that keep together always fly into the fence. I think the answer must have to be somehow related to momentum. At first I thought this couldn’t make any sense. A derivative of Newton’s laws proves that momentum is conserved. M1V1 = M2V2. But then I realized, we’re again considering TOTAL momentum. Like the example with the shotgun and the cannonball, total momentum of the entire spray is one thing, however, the momentum of each pellet in the spray is much less. After heading down the hill as a group, there is a given momentum. Letting go of each other’s hand DOES NOT change the TOTAL amount of momentum. However, now the momentum of each individual is much less. How this effects the process of slowing down has to do with friction. The momentum difference let's the larger group to surpass the friction. At first this never made sense to me. I reasoned that if two objects of different mass were pushed, the momentum of the larger object would not allow it to move farther. The smaller object would move farther, because that’s just common sense. But I realized two flaws in my thinking. First of all, if two objects of two different masses are moving at the SAME velocity, the forces must be different. If I am pushing a larger object at a greater force than that of a small object, and both objects are at the same speed, it takes a much larger opposing force to slow the larger object down. And since the friction at the end hill isn’t very high, the large group’s momentum carries it into the fence. Think about throwing a tennis ball compared to an air filled balloon. If both were moving at the same velocity, and were allowed to slow down, the tennis ball would fly farther. Now imagine a normal tennis ball, and the tennis ball from House ( I luv him <3!). If both are travel at the same VELOCITY, the larger one will fly farther if they are both allowed to slow down. Hope that clears things up, this question’s been bothering me for months. It’s great to know there are other people who wonder about the same kinda things as me.

 

[khkwang]

Hey, Laura that is quite the co-incidence haha. I guess people like us can't help but wonder "why?" when we see something that doesn't make sense to us.

I was reading over your argument, but I'm going to have to disagree with your argument.

force of gravity
it's true that the force of gravity would be different between the two, but assuming that gravity is the only force acting on the tubers,

F_net = Fg =>> ma = mg =>> a = g

So even though the force of gravity is greater for the group, the acceleration is still the same regardless of mass (which makes sense, all masses fall at the same acceleration when neglecting air resistance).

momentum
It's also true that the momentum of the group is larger than that of the single rider, but even so, the velocity of the group and of the singe rider would be equal after separating.

friction
Assuming that friction is now the only force acting on an object,

F_net = F_f =>> ma = -mgu =>> a = -gu

In which the acceleration on the object is once again independent on the mass of the object.

air resistance
It's true that each individual rider will have a lower force of air resistance on them, but since they are a smaller mass, the amount of air resistance needed to slow them down would be smaller as well, than as a group.

Lets assume at a particular snapshot in motion, the force of air resistance on a single rider is 3N.

So for the single rider,
F_net = F_air = -3 =>> ma = -3 =>> a = -3/m

Now imagine a group of 3 riders, with one of the riders being in the back, and thus is hidden from air resistance. Then the force of air resistance only acts on 2 of the riders, and as a whole, equals -6N. Then
F_net = F_air = -6 =>> Ma = -6 =>> a = -6/M

Assuming that all three riders are equal in weight, M = 3m and so for the group,
a = -2/m

As you can see, the deceleration due to air resistance on the group is smaller in magnitude than than on the individual rider, and thus explains (or at least partially explains) why the individual slows down faster than the group.

your final argument about momentum
You said that "it takes a much larger opposing force to slow the larger object down", which is true. But since the force of friction is proportional to the mass of an object, the force of friction on a larger object is actually larger than the force on a smaller object, even though they're sliding on the same surface. The mass from momentum (p = mv) gets canceled out by the mass in the force of friction ( F_f = -mgu) and so the resulting deceleration on the larger object, and the deceleration on the smaller object are actually equal. So generally, one could say that the deceleration due to friction is independent on the object's mass.

 

[Laura681]

I KNOW LOL! That’s what I told my boyfriend! Those exact things made all those points make no sense. We’re both complete nerds, and he’s kind of anal so we kept on debating about it. But then he said something that made me think one eighty about everything. I can’t use Newtonian physics by itself to reason this out because it just isn’t possible to do all the calculations. Firstly, all the assumptions made by Newtonian physics don’t apply. We aren’t working with perfect conditions, so formulas on paper don’t apply one hundred percent. The hill isn’t actually a uniform slide. Friction is therefore different all over the place. The sketches we make in the ice are made by rakes, and they aren’t exactly perfect. Also when people have their hands joined in a group, that’s not the same as one object. So it isn’t simply just the comparison of a large thing and a small thing. I know mass isn’t actually a factor involved with friction, however I think it is if pressure is kept constant. Because joining hands in a group isn’t really like having one object, by the base of the hill pressure might actually stay relatively constant… I don’t know. Something like that. I’ve been trying to make sense of this for a while. I hate my bf for it… but I think he might actually have a bit of a point about the limits of the practicality of physics theory. People aren’t the same mass, so spins are kind of elliptical. The way people are sitting need to be factored in. The shape of the tube even needs consideration. There are too many factors to be considered to come to any actual conclusion. There has to be a cooperative effort between many forces that slows the riders down. Air resistance, friction, momentum, possibly pressure differences are all probable factors that slow the single riders down. Also, your idea of air pressure being proportionately reduced is true, but not 100% practical. There’s no way in the real world that air resistance would slow anybody down enough. Also, air resistance is blocked by the people in front but only partially. In a group of three, yes the two in front would give the guy at the back a bit of a tail-wind effect, but the gaps between the tubes would leave a lot of air still directly being funneled towards the back person. The net result would be a slight decrease of air pressure proportionately, but the difference would not be significant to stop the riders.

All this talking has confused me again… I made my peace with friction and momentum. But now I need to think everything over. I still think it has to do with friction and momentum, maybe something to do with angular momentum too caused by spin, the only problem now is why. I’m thinking riders who separate spin more frequently. Spinning increases friction, and increases heat, Heat causes escape of energy, some of the kinetic energy escapes as heat. Kinetic goes down, so momentum goes down. I think it might actually be an exponential situation here. I’m not sure what it is that slows single riders down better than large groups but its been fun either way. I love nerding out in private, it’s a great release. I’m usually talking about stupid girl stuff with my roommies and I can’t really be myself. My boyfriend just gets annoyed after a while. Although, I think, without doing like a hundred different calculations, you’ll never get to the end of this, so we can only try understanding the basics. All we know is that there is a difference in mass and surface area. The only things in Newton’s phys that is effected by mass is gravity, possibly friction (because we don’t really know if the formation of the circle can be treated like an actual proportional object. Friction is not affected by mass only if pressure is changes proportionately. We cannot say whether or not pressure changes in the way you would expect a solid object in this situation), and pressure. There’s probably a few more that I can’t think of right now. Whatever the reason, keep posting, I’d like to hear it. But I don’t know how well calculating physics will hold up unless you do every calculation possible. Like that guy from Numbers. LOL I wonder if that’s actually possible. You should try it! LOLOL!

 

[khkwang]

hey laura sorry for the late reply, but I was hoping someone else would chime in with an idea.

Anyway, I can't get myself to agree with you. But I also can't get myself to disagree with you either.

The stubborn part of me that believes that equations (if all factors are considered) should explain a given system extremely accurately (at least at this scale). If this weren't true, then we wouldn't have much of the technology we have today. Engineers don't just go "Well this should work... I think".

I think you're right about air resistance. It doesn't see to be a strong enough force to be the major factor here.

I still have my doubts about friction though, but I'm starting to rethink momentum, specifically angular momentum.

When you're all joined as one, everyone's legs are in the middle, concentrating the mass more in the middle and so the group keeps spinning. But once everyone let's go, individual legs flail about, and destroys the symmetry and the spinning slows down. I think the slowing down of the spinning "steals" momentum from forward momentum.

...There's not really much mathematical basis on what I'm saying, and I'm pretty sure it doesn't even make sense when you actually try to work it out, but that's the last theory I got. Maybe you can test it out by instructing a kid to keep his legs in the tube to keep the circular symmetry and report back here with the results? lol

 

[Zula110100100]

Spin makes the most sense, I think. But, Possibly the separated peoples trajectories would diverge into a less vertical pattern? Are the people more side by side or inline? Would conservation of angular momentum imply that a spinning object breaking up would result in smaller objects spinning faster as well? Or would it be transferred into different linear velocities, i.e. in a clockwise spin the one on the right side(watching from behind) would have a lower linear velocity and the one on the left would have a higher linear velocity...not sure how that would work

 

[Pythagorean]

Is it possible that collisions within the group at the end would propel the people in the front of the group forward and people in the back of the group would slow down more so that momentum is still conserved, but the people in the front of the group are going to hit the fence and the people in the back of the group will fall shorter than they usually do?

 

[willem2]

The people in the back probably have less friction, because the people in front make the snow/ice smoother.