Conservation of Momentum Question

[crddrc]

Hello,

Here is the situation I have been pondering:

A person is sitting in a shopping cart at rest. By throwing their weight forward they are able to cause the shopping cart with them inside to move. My question is how this works. If momentum is conserved unless an external force acts on the system and the initial momentum is zero when the cart is at rest, how is the shopping cart with the person inside able to move forward?

Thanks

 

[K^2]

If the cart is completely frictionless, it won't work. The center of mass will stay put.

In the real world, however, there is going to be friction. You can use static friction to allow you to pick up the speed without pushing the card back. Effectively, via friction of cart with ground, you are pushing from the ground.

Once you pick up speed, you hit the wall of the cart. The impact generates brief, but very high force. That force easily overcomes static friction causing the cart to roll. Once in motion, friction is much lower, so you can travel some distance before coming to a stop.

In terms of conservation of momentum, the momentum of you + cart is not conserved, but that's not a closed system due to friction. Momentum of you + cart + Earth is conserved.

 

[Simon Bridge]

Welcome to PF;
Notice that the cart moves in the direction the person throws themselves? Throwing forward should make the cart go backwards shouldn't it? That should be a clue... there is friction involved.

It is the sudden shift from going forward to stopping that overcomes the static friction, allowing the cart to move. The change in momentum in that short time is backwards, so the cart goes forwards. (You may not think of this as an "impact" ;) )

 

[crddrc]

Thank you Simon and K^2! Great answers from both of you.

 

[cragar]

If the cart is completely frictionless, it won't work. The center of mass will stay put.

Is this similar to sitting in the cart with a paint ball gun and shooting the front of the cart.
Because I thought if we were shooting the front of the cart it would go forward because the change in momentum is greater when the ball bounces off, even with no friction.
Or I guess kicking might be different, I am probably way off.

 

[jbriggs444]

Is this similar to sitting in the cart with a paint ball gun and shooting the front of the cart.
Because I thought if we were shooting the front of the cart it would go forward because the change in momentum is greater when the ball bounces off, even with no friction.
Or I guess kicking might be different, I am probably way off.

[Assuming the wheels are frictionless...]

If the paint ball lands inside the cart then momentum is conserved and you end up motionless.

If the paint ball lands outside the cart then what you have is essentially a needlessly complex rocket motor.

 

[Matterwave]

I should mention that in a frictionless case, the cart would move backwards while you were running forward and then perhaps go forward a bit when you hit the front, leaving the COM at the same place. But because running forward provides a much smaller force, the cart does not overcome static friction until you hit the front of the car.

The possibility of moving the car forward while you're in the car is the same as the possibility of moving yourself forward while standing without anybody pushing you.

 

[Buckleymanor]

[Assuming the wheels are frictionless...]

If the paint ball lands inside the cart then momentum is conserved and you end up motionless.

If the paint ball lands outside the cart then what you have is essentially a needlessly complex rocket motor.

Figueres if that's the case, would a bow and arrow and a shortish piece of string attached to the bow and the arrow be similar to the OP question.
You sit in the cart and fire the arrow.
Would the cart move backwards when you fire.
Would the cart move forewards when the arrow reached the end of it's flight and jolted the string, the bow, the person, the cart.

 

[K^2]

Figueres if that's the case, would a bow and arrow and a shortish piece of string attached to the bow and the arrow be similar to the OP question.
You sit in the cart and fire the arrow.
Would the cart move backwards when you fire.
Would the cart move forewards when the arrow reached the end of it's flight and jolted the string, the bow, the person, the cart.

Frictionless case? The cart would recoil and start moving backwards when you fire the arrow. The cart and arrow would come to a stop once the string is fully extended. The center of mass would remain stationary throughout the experiment.

 

[Simon Bridge]

... when you reel in the arrow, slowly, the cart stays in place. Reel in the arrow quickly and you risk overcoming the static friction.

 

[A.T.]

Aside from the static friction that prevents rolling, there is also the lateral static friction that prevents wheels from sliding sideways. Since a shopping cart has caster wheels this also plays a role when someone is throwing his weight around in the cart. Even with wheels that have zero static rolling friction, you could potentially get the center of mass moving.

 

[Buckleymanor]

Frictionless case? The cart would recoil and start moving backwards when you fire the arrow. The cart and arrow would come to a stop once the string is fully extended. The center of mass would remain stationary throughout the experiment.

Why would you start moving backwards you the bow and the cart would recoil but I don't imagine it's in that direction.

Bow Recoil - AKA, Hand-Shock

Some call it kick, or hand-shock, or refer to it as shot-vibration, but we're all usually referring to the same thing, recoil. Of course, a bow's recoil is rather backwards from that of a gun - pushing away instead of towards you. But the phenomenon is basically the same - an undesirable jolt at the point of the shot. Why does it happen? It's Sir Isaac Newton's fault of course. When a bow is drawn, the limbs compress back under tension. When the bow is fired, the unloading limbs jolt forward and return to their original positions. Since the cams are attached to the bow's riser, the inertia of the fast-moving limbs (Limb Thrust) causes the bow's riser to jump forward too. And since your hand is attached to the riser at the bow's grip, you feel the riser's abrupt movement as recoil. It's a natural byproduct of such an explosive energy release, and on some bow designs it's quite noticeable - perhaps even detrimental

 

[K^2]

Why would you start moving backwards you the bow and the cart would recoil but I don't imagine it's in that direction.

The movement of limbs actually results in two recoils. When the limbs start to accelerate forward to propel the arrow, the bow actually pushes towards you, but it's a fairly gentle push over longer time. When limbs stop, it's rather sudden because of the string, so you get the kick that the quote talks about. It's particularly severe on recurve bows simply due to the geometry of the limbs. You feel that kick a lot more because of how sudden it is, but the actual momentum transfer is smaller, because the initial push contains both momentum transferred to the limbs and to the arrow, and the arrow's momentum is not recovered.

It's the net difference due to the arrow's momentum that results in your total momentum after firing the arrow being directed backwards. So the cart will roll backwards as result of the recoil until the arrow is stopped by the rope tied to it.

Edit: This would actually be a fun one to demonstrate like that. If the weather improves for a day or two, I'll try to film it. The momentum carried by the arrow is quite significant. Should be enough to get me rolling on a skate board.

 

[Buckleymanor]

The movement of limbs actually results in two recoils. When the limbs start to accelerate forward to propel the arrow, the bow actually pushes towards you, but it's a fairly gentle push over longer time.

Not so sure that the bow pushes much towards you once you let the arrow go. It's in tension as you draw the bow and is pushing towards you as you hold it.Once you let go it can only move towards you with the same force as the arrow and string is moveing in the opposite direction.In all there must be three recoils two in one direction and one in the other.
My bet is the two recoils has more force than the one.

Edit: This would actually be a fun one to demonstrate like that. If the weather improves for a day or two, I'll try to film it. The momentum carried by the arrow is quite significant. Should be enough to get me rolling on a skate board.

Go for it and let us know the results.

 

[jbriggs444]

Not so sure that the bow pushes much towards you once you let the arrow go. It's in tension as you draw the bow and is pushing towards you as you hold it.Once you let go it can only move towards you with the same force as the arrow and string is moveing in the opposite direction.In all there must be three recoils two in one direction and one in the other.
My bet is the two recoils has more force than the one.

Since momentum is strictly conserved and the arrow ends up at rest, my bet is that the impulse delivered by all of the recoils plus the tug on the string attached to the arrow sums to zero.

 

[K^2]

Not so sure that the bow pushes much towards you once you let the arrow go. It's in tension as you draw the bow and is pushing towards you as you hold it.Once you let go it can only move towards you with the same force as the arrow and string is moveing in the opposite direction.In all there must be three recoils two in one direction and one in the other.
My bet is the two recoils has more force than the one.

It is basically your guess against conservation of momentum. Doesn't look good for your bet. :)

But I've got nothing against the experiment. So I'll set it up. I've figured out location, so now I just need to borrow a skate board from somebody. (Mine's a 2-wheel caster, so that won't work.)

Now, what would be really great is if you could see the influence of all the individual recoil stages, but I doubt I'd be able to catch that on camera. The net result should be plainly visible, though.

 

[cragar]

[Assuming the wheels are frictionless...]

If the paint ball lands inside the cart then momentum is conserved and you end up motionless.

If the paint ball lands outside the cart then what you have is essentially a needlessly complex rocket motor.

ok when I fire the paint ball into the front of the cart it will move forward right
becuse the change in momentum is greater when it bounces of the front of the cart .
ok so now it is headed to the back of the cart and it bounce of their so that should appose the motion. Now its headed back to the front of the cart. it still seems like the cart will move forward on average. There is porbably some subtle things I am missing.

 

[rcgldr]

momentum is conserved unless an external force acts on the system and the initial momentum is zero when the cart is at rest, how is the shopping cart with the person inside able to move forward?

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]

ok when I fire the paint ball into the front of the cart it will move forward right
becuse the change in momentum is greater when it bounces of the front of the cart .
ok so now it is headed to the back of the cart and it bounce of their so that should appose the motion. Now its headed back to the front of the cart. it still seems like the cart will move forward on average. There is porbably some subtle things I am missing.

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 \Delta p = \lim_{n \to \infty} (-1)^{(n+1)}a^nmv = 0.

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]

ok when I fire the paint ball into the front of the cart it will move forward right
becuse the change in momentum is greater when it bounces of the front of the cart .
ok so now it is headed to the back of the cart and it bounce of their so that should appose the motion. Now its headed back to the front of the cart. it still seems like the cart will move forward on average. There is porbably some subtle things I am missing.

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]

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.

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 piece 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]

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]

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.

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.