Can you verify if there is a error in the notes or i am just stupid.

[jessicaw]

as uploaded.
I think the example is very weird.
Why we can use conservation of momentum? There is external force!

thx.

 

[Doc Al]

The assumption is that the tension is the same throughout the string. So, in effect, the two masses are exerting equal and opposite forces on each other and thus momentum is conserved. Since the impulse is of short duration, any external forces on the masses can be ignored.

 

[kcdodd]

I think strictly speaking, jessicaw is being diligent. Momentum is clearly not conserved between the two masses in "normal" coordinates. The forces are not equal and opposite, they're just equal both in the up direction. What it is really assuming is that the two masses will have the same change of momentum; not that momentum is conserved. i.e pf - pi = (-mV) - (-mv) = (MV) - (0); because M has V and m has -V (-v initially).

 

[Doc Al]

I think strictly speaking, jessicaw is being diligent. Momentum is clearly not conserved between the two masses in "normal" coordinates. The forces are not equal and opposite, they're just equal both in the up direction. What it is really assuming is that they will have the same change of momentum; not that momentum is conserved. i.e pf - pi = (-mV) - (-mv) = (MV) - (0); because M has V and m has -V (-v initially).

I completely agree. It's not something 'obvious', but something that requires explanation. I think it was a good question that the textbook failed to properly address.

When I said the forces are in effect 'equal and opposite', I was of course taking my coordinates along the rope. (That too, should have been explained. )

 

[D H]

I think strictly speaking, jessicaw is being diligent. Momentum is clearly not conserved between the two masses in "normal" coordinates. The forces are not equal and opposite, they're just equal both in the up direction.

Correct. As you noted, the forces are not equal but opposite. They are equal and non-opposite. So there should be no expectation that momentum is conserved (and it is not conserved). One way to resolve this is to come up with an analog that simplifies and unfolds the system so that the forces are equal but opposite. The attached explanation did not do that, however.

Edit
Just to make things clear, the result obtained in the attachment to the original post is correct. The issue is the handwave regarding conservation of momentum needs a bit better explanation and justification.