Why Is Momentum Conserved in a Variable Mass System Like a Sand-Laden Cart?

[Eitan Levy]

Homework Statement

A cart with no motor moves on a plane. At t=0 it has a mass equal to m₀ and some velocity. Each second sand with a mass of α comes out of the cart with a velocity of 0 in relation to the plane. What is the equation of motion of the cart?

Answer: dV/dt=αv(t)/(m₀-αt)

Homework Equations

F=dP/dt

The Attempt at a Solution

Basically I can't understand why the momentum of cart is conserved. I mean, if sand comes out of it with a velocity that is different than the velocity of the cart there has to be some force involved right?

What I get is dV/dt=2αv(t)/(m₀-αt), because the loss of momentum of the sand translates into some force that the cart applies on it (I believe). What am I doing wrong?

 

[Adhruth Ganesh]

Basically I can't understand why the momentum of cart is conserved. I mean, if sand comes out of it with a velocity that is different than the velocity of the cart there has to be some force involved right?

No force will affect the motion of the cart because the sand particles fall vertically downwards while the cart moves in a horizontal direction. Thus, there is no force which affects the linear momentum of the cart.

 

[Eitan Levy]

No force will affect the motion of the cart because the sand particles fall vertically downwards while the cart moves in a horizontal direction. Thus, there is no force which affects the linear momentum of the cart.

But what causes the sand to suddenly lose its momentum?

 

[Orodruin]

There is an internal force in the sand-card system. However, since the sand momentum is zero, conservation of momentum in the sand-cart system therefore requires that the momentum of the cart remains the same. Note that the momentum of just the cart is actually increasing because some of that momentum was previously carried by the sand.

 

[jbriggs444]

But what causes the sand to suddenly lose its momentum?

The problem says that it does. If you wish, you can imagine a gremlin standing on the cart and carefully throwing grains of sand rearward just hard enough so that they become motionless with respect to the ground beneath.

 

[haruspex]

No force will affect the motion of the cart because the sand particles fall vertically downwards

How did the sand go from moving at the same velocity as the cart to falling vertically (in the rest frame)?