Conservation of Momentum Question

rcgldr

If you only consider the cart, then the external force is geneated by the earth in reaction to the internal force which is transmitted to the earth via friction between the wheels and the earth (Newton third law pair of forces). Even if the wheels are aligned in the direction of movement, friction in the axles of the wheels is enough for a small reaction force related to the person moving within the cart to be applied over a longer period of time, then overcome by a sudden jerk by the person in the opposite direction.

If you consider the cart and earth as the closed system, with the car initially at rest with respect to the earth, then the center of mass of cart, person, and earth does not move, regardless of any relative movement between cart and earth, and momentum of this system is conserved.

Simon Bridge

Conservation of momentum says that the recoil has to push the cart backwards - the bet here is whether the initial backwards push is better than friction in the wheels?

K^2

This is a bit difficult to set up with cart that's also moving, but if we assume that cart is "infinitely heavy", you can do an estimate.

Say, after each bounce, velocity goes from v to -av for some a 0<a<1. Technically, we'll get infinitely many bounces in this approximation. But this kind of infinite sum we can still do.

So first, the ball is fired, and cart receives -mv. When the ball hits front wall, ball recoils with -av, and cart receives (a+1)mv. Now the ball bonces off from the back wall, boing +a²v now, and cart picks up -(a²+a)mv.

Total transfer: Δp = -mv + (a+1)mv - (a²+a)mv + (a³+a²)mv - ...

Rearranging terms slightly: Δp = -mv + mv + amv - amv + a²mv - a²mv + a³mv - ...

So every term cancels at except for the last one. After infinitely many bounces [itex]\Delta p = \lim_{n \to \infty} (-1)^{(n+1)}a^nmv = 0[/itex].

The shortcut, of course, is to note that if the final velocity of the ball relative to cart is zreo, then so was the total momentum transfer. Of course, like other people have said, friction can make a difference.


P.S. I've tracked down a skateboard I can borrow and will set everything up in the next couple of days. I'll try to see if I can manage to arrest the arrow with a string as well, but it might be a bit difficult to achieve.

Buckleymanor

The subtle thing is probably is the first recoil.
When you first fire the paint ball the gun recoils so it moves backward's first then foreward's when it hits the front.
The reasoning is correct but in the wrong direction and gets less over time.

K^2

So that was a bust. Recoil from arrows is nowhere near enough to get the skateboard to move with me on it. And that's with me being 145lb, firing aluminum arrows from 5' 30# recurve bow. I gave it about as good a chance to work as it can get. Nada.

The only way I can see to reduce friction enough to get an effect at this point is suspending myself by the climbing harness from a long rope. I know just the place to do this, but the weather is no longer cooperative around here. If something like this will come up again in the late spring or summer, I'll definitely try it.

halo31

I would think the paintball example is just a simple example on an inelastic collision where two objects collide with each other and they stick together when they reach their final position.

Buckleymanor

You don't have to be on the skateboard if you wan't to reduce friction.If only you could find a way to trigger the bow or borrow a crossbow and place it on the skateboard and fire it with a peice of string or other device on the trigger.
A crossbow suspended on it's own on a rope and fired would probably tell us what's likely to happen.

K^2

There are plenty of mechanisms I can build that demonstrate conservation of momentum. But it's been done before plenty of times. Showing an actual person get actual recoil from an actual bow, that would be kind of interesting. I don't really see how a crossbow on a skateboard would be any better than demos they show on air track.

johns123

So I went out and tried it on my ice skates. If I leaned forward slowly, nothing happened. But if I leaned forward quickly, I moved forward quite well. If I added a squat to the quick lean forward, I went forward even better. I think I notice that the move forward actually starts when my lean forward quickly stops. Also, the squat action brings my skates back under me. Friction on the ice is not much, so I don't think I'm pushing against the ice. I think my quick movement is pushing against my own inertia that is holding me still ( is that an Impulse .. Ft = -mv where the F = mv/t = ma ? ).

johns123

Hmm .. I'm also a springboard diver. If I spring into the air, and then want to do a full twist ( spin about my body axis ), I quickly swing my arms in the direction I want the spin to go .. and it goes that way. Also it seems to have nothing to do with pulling in. That only controlls the speed of the spin. Again, I think I'm working with acceleration change rather than just mass - velocity. I don't understand it. It is just a fact. The "quick" factor seems to be the cause ??

K^2

With ice skates, it's still about friction. Keep in mind that friction under the skates can vary a lot depending on the film of water forming underneath, and that depends on the pressure. If you suddenly reduce pressure, the water can actually freeze and briefly cause the skate to "stick" to ice. This would be consistent with effect amplified by you squatting down. I also don't know how well you managed to keep your skates parallel. Even a tiny angle can let you propel yourself by exploiting lateral friction, which is quite high.

With rotations, things are a bit more complicated. You CAN change your orientation in space without expelling anything. It gets pretty complicated in terms of physics because there are so many degrees of freedom, but it's basically what cats do to land feet-first. And yes, you can use it to adjust your orientation in a dive.

johns123

[QUOTE=K^2;4169793]With ice skates, it's still about friction.

I'm sure you are right, but the real world problems are a nightmare to understand. What about my diving spin. There's no friction there, but I spin in the direction I throw my arms. Also, if I just move my arms in that direction, my body goes in the other direction, and I don't spin at all. I was once told that it is 2 problems. First I twist in the direction I want to go .. it stops .. and then I go in that direction.

Buckleymanor

Not seen air track demos I will have a look.
The mass of the crossbow would act like a person(substitution) a lot lighter in proportion and be still.In effect it would be more idealised and likely to work because of the weight ratio of the crossbow to the force of the arrow and the frictional forces of the skateboard.

Anan275

Does anyone know how do this question:
1. Student 1 pulls to the left with a horizontal force on a 60 kg crate on a smooth floor. Student 2 pulls to the right on the same crate with a force of 250 N at an angle of 40º above the horizontal. The crate starts from rest and when it has moved 20 m [right] it has a velocity of 4.2 m/s [right]. Find the work done by each student on the crate.

K^2

Like I said, rotation is way more complicated. Mass of an object cannot change, except by splitting into several objects. That's why you are so limited when dealing with conservation of momentum. Both total momentum and total mass of the system are fixed, so you can't do anything about velocity of the center of mass.

With rotation, the equivalent is the moment of inertia. And moment of inertia of a body depends on configuration. That's why your spin accelerates when you pull in the limbs. But it's also why you can turn your body seemingly by throwing your weight around. What you are really doing is turning one way and then another, but from different configuration, so as to have different moment of inertia. The result is a net rotation which could not be achieved otherwise.

I know it might feel this way, because this sort of movement is very intuitive to human brain, but you do a lot more than just throw your arms in a particular direction. Your entire body is involved in making that direction change.

Simon Bridge

@Anan275: that would be off-topic for this thread, which is about conservation of momentum. You should start a new thread for that question.

Buckleymanor

I have had a look at air track and I am at loss to some of the explanations with regards to conservation of momentum and the mechanism in question( bow crossbow arrow string).
If you could point me towards an air track vid of the effect it would help.
In the meantime could you explain if it's an elastic, inalastic, or completly inelastic collision.
I go for the completly though I am not absolutely sure of my reasons and this is why the questions.