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A cart filled with a mass $m_0$ of sand leaks it at a constant rate of $\mu \ \mathrm{kg/s}$ through a hole at the bottom of the cart. Suppose that the cart is moving on rails. Find the normal reaction force from the rails as a function of time. Ignore the mass of the empty cart.

Does the function $\mathbf{N}(t) = (m_0 - \mu t) g\hat{\mathbf{j}} $ really work? Doesn't the sand leaving through the hole give an upthrust to the cart, to reduce the effective normal weight? Note that the sand gains momentum in the $y$-axis.

I'm feeling confused at this point.

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    $\begingroup$ What makes you confused? That this looks, a little bit, like a rocket? OK, so let's reason trough that. What is the velocity of the sand at the point where it leaves the cart? Zero, right? What's the thrust of a rocket motor that expels propellant at a velocity of zero? $\endgroup$
    – CuriousOne
    Commented Aug 30, 2014 at 13:38

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Yes - it really is that simple. There is no "upthrust" from the sand falling out of the car. There is just less mass in the car, and thus less force of gravity on the car. The sand gains momentum because gravity continues to pull on it after it leaves the car - but that no longer affects the car.

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