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In Kleppner and Kolenkow's 1st edition book, there was an example question : Sand falls from a stationary hopper into a freight car moving with uniform velocity v. The sand falls at the rate dm/dt. Find the force required to keep the car moving.

Since it's an example the solution is given which is $v*dm/dt$ force is required. However, in the end it also writes," We can understand why this force is required considering in detail just what happens to the sand grains when it lands on the surface of the freight car. What would happen if the surface of the freight car were slippery?"

But I really couldn't think of the force acting on the sand or what would happen if the sand grains fall on slippery surface.

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  • $\begingroup$ Note that the question and solution are inconsistent: the sand falls "into" a freight car, but in the solution, it "lands on the surface" of the car... $\endgroup$
    – DJohnM
    Commented Apr 3, 2018 at 21:13
  • $\begingroup$ @DJohnM The freight car initially was empty before coming under the hopper. Maybe it's the initial condition. $\endgroup$
    – suiz
    Commented Apr 4, 2018 at 7:34

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The freight car has a horizontal velocity the grains of sand have zero horizontal velocity (momentum).
To accelerate the grains of sand to the horizontal velocity of the freight car a horizontal force must act on the grains of sand.
The origin of the horizontal force is kinetic friction between surfaces (falling sand to sand already moving with freight car, falling sand to freight car) which are moving relative to one another.

I think the quote is asking you as to what happens if there is no frictional force; the answering being that a heap of stationary sand builds up under the hopper.

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  • $\begingroup$ So can I think of it analogous to the condition when I am to land on a super slippery ship (frictionless), me being a point object, the ship will move beneath me with constant velocity, while I would be were I was? (I'm viewing myself like a sand grain.) $\endgroup$
    – suiz
    Commented Apr 4, 2018 at 7:41
  • $\begingroup$ @suiz Yes, I think that is what the authors were suggesting. $\endgroup$
    – Farcher
    Commented Apr 4, 2018 at 7:42
  • $\begingroup$ And hence no force needs to be applied at all for the frictionless condition. When there was frictional force between the surface and the sand the friction between them would act in opposite direction freight car and thus, slow it down. Hence the force we apply overcomes the frictional force acting on the car due to the falling sand. Do you think this can be an appropriate explanation for the results obtained? $\endgroup$
    – suiz
    Commented Apr 4, 2018 at 7:48
  • $\begingroup$ @suiz If no force is applied the freight car will slow down. The force applied to the freight car does work and that work increases the kinetic energy of the sand and also produces heat due to there being relative movement between the sand grains and the freight car. $\endgroup$
    – Farcher
    Commented Apr 4, 2018 at 7:51
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The force required can be calculated from change in momentum. Sand velocity must eventually match velocity of the train (unless train has no rear wall and it just falls off the back). So slipperiness or coefficient of friction is irrelevant. All momentum must be transferred to the train.

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