# Trouble conceptualizing Conservation of Momentum

I just learnt the concept of conservation of momentum, and wanted to make sure my thinking was correct on a toy example.

## Question

Suppose you're in a car full of frictionless sand traveling on a frictionless road at constant speed $$v$$. A hole suddenly appears in in the bottom of the car, and the sand starts pouring out. My question is does the car speed up as the sand pours out?

## Reasoning

My thinking is that for the the system of (myself, the car, and all the sand) the momentum is not conserved, because the sand that falls out gains vertical momentum from falling.

If we choose the system as (myself, the car, and the sand still in the car), the momentum is conserved, because the external forces in the vertical direction (the normal forces and the weights) cancel out, and there are no external forces in the horizontal direction.

Therefore, we have the initial momentum equalling the final momentum: let $$m$$ be my mass, let $$m_c$$ be the mass of the car, and let $$m_s$$ be the mass of the sand in the car still remaining.

We have $$(m + m_c + m_s)v = (m + m_c + m_s)v_f,$$ so clearly $$v_f = v$$.

Is this logic correct? Would love some tips/clarification on picking the system and determining the momentum if not.

• The leaking cart is a bit more complicated than that. See physics.stackexchange.com/q/1683/25301, physics.stackexchange.com/q/659460/25301 along with the inverse problems physics.stackexchange.com/q/699160/25301, physics.stackexchange.com/q/153361/25301 and so on Commented Aug 24, 2023 at 17:09
• @KyleKanos Very interesting. If I simplify the problem so that the sand doesn't leak out, but the cylinder of sand above the hole all falls out at once, leaving the rest of the sand around the hole untouched, we do have that the velocity stays the same, yes? And thanks for the response! Commented Aug 24, 2023 at 17:23
• An important feature of momentum, unlike energy, is that it is a vector quantity - if we look at the vertical and horizontal directions (as you did), there is horizontal momentum and vertical momentum, and they are separate. The sand gaining vertical momentum, as you noted, will not change the horizontal momentum (and thus the horizontal velocity) at all. But as Kyle noted above, this problem is more complicated - the sand not only moves vertically out of the hole - the sand in the container will also shift horizontally to "even itself out", and can change the horizontal momentum. Commented Aug 24, 2023 at 17:39
• @Nadav Har'El Great explanation, thank you! Commented Aug 24, 2023 at 17:55
• @PhysicsNoob In either case (slow leak or fast leak), you have a variable mass system in which the conservation of momentum must be handled very carefully. Commented Aug 24, 2023 at 18:02

The sand falling has zero vertical momentum when it leaves the car, so no vertical momentum change occurs. The sand does have horizontal momentum, so the horizontal momentum of the car plus the remaining sand changes. However this change is purely due to the change of the mass and the velocity remains constant.

I cheated for the vertical momentum. The car is vertically accelerated by gravity and the reaction force of Earth. As the falling sand does not feel the reaction force, there is a reduction of reaction force of the car plus the remaining sand, exactly compensating the reduction in gravity. Now you may take the tyre pressure and the suspension system in consideration, and the spring constant of the tarmac. These combined springs relax ignorer to reduce the reaction force so the car will move up a little. Unless is is a hydraulically suspended one.

Let's ignore the centrifugal force due to Earth ...

Your equation makes no sense. Is $$m_s$$ the same number at both sides? Is the left side of the equation a situation at different time than the right side? If so, then the $$m_s$$ should be a different number at the left side and the right side.

Anyway, there is nothing to calculate.

Think like this: There is a car. Is there momentum transfer between the car and something else? No. So the car does not accelerate.

By 'car' I mean a thing consisting of wheels, motor and chassis.

• Is there momentum transfer between the car and something else? No. So the car does not accelerate. I suggest you read through the links I posted as a comment to the original post as this block I've quoted is definitely wrong. Commented Aug 24, 2023 at 21:00
• @KyleKanos "The water jet from the nozzle is directed vertically down." The sand flow is NOT directed vertically down, as there is no adjustable nozzle in the car. Seems like two very different things to me. Did not read everything. Commented Aug 25, 2023 at 3:07
• From OP: A hole suddenly appears in in the bottom of the car, and the sand starts pouring out If the bottom is not pointed vertically down, then we have very different understandings of either vertical or bottom. Either way, as I commented to the OP: you have to be very careful in how you treat conservation of momentum. This is because the mass that is being lost (the sand) actually contributes to a slight gain in the speed of the car ($\propto v\dot{m}$)! Commented Aug 25, 2023 at 10:59