Changing momentum of moving trolley Consider a trolley of mass $m$  moving at a velocity $v$ along a smooth horizontal plane. 
It is full of water, and water is leaking at a constant rate out of the bottom of the trolley, i.e perpendicular to the plane. 
My questions are: what happens to the momentum and the kinetic energy of the constituents of this system (water and trolley)?
It can be said that as the trolley's mass decreases, its velocity will increase as implied by the law of conservation of momentum - However, since the water is flowing out perpendicular to the plane, it is not in the same linear direction as the trolley - so velocity would stay constant, mass would decrease and so the momentum of the trolley decreases. I'm not quite sure which one of the two are correct.
In addition, since mass decreases, but K.E. = $1/2$mv$^2$ if the velocity increases then the KE should increase. however, if velocity remains constant, K.E will decrease. Can someone help me discern which one will occur?
 A: I think that your problem is that you are only taking the trolley into account when applying the law of conservation of momentum. Conservation of momentum considers the whole system, and the water is still part of the system even when it falls out of the trolley. In fact the ground also becomes part of the system when the water collides with the ground. The trolley would not gain velocity when the water falls out, because the water would bring some of the momentum with it, proportional to its mass, considering that it has equal velocity to the trolley. If it didn't have equal velocity to the trolley, that would imply that there was an external impulse such as air resistance acting, which voids the law of conservation of momentum anyways. When the water hits the ground momentum is still conserved. The water slows down, but the earth - which is part of the system, actually DOES accelerate. The earth gets moved by the collision of the water, but just by an amount that is not noticeable, because the earth is so massive.
So basically KE and Momentum are both conserved if you are looking at the entire system, which includes the trolley and the water that has fallen out (assuming that the water collides elastically with the ground and the ground is included in the system, or it never hits the ground.) But looking just at the trolley, KE and Momentum are lost in direct proportion to the mass of the water that falls out, and velocity of the trolley remains unchanged.
A: Notice that water and trolley are part of a system. If water is flowing out at a constant rate (or not) its momentum will be changed only by an external force, say friction with the floor when the water reaches it. If you consider the trolley as just the trolley, its momentum and kinetic energy won't change. If you consider the system as a whole you need to ask yourself what happens with the water after it exits the trolley: for example, it is being accelerated by gravity, therefore its momentum is changing.
A: "velocity would stay constant, mass would decrease and so the momentum of the trolley decreases" , is a correct argument , but if this is a system where gravitational forces are present , you have neglected the pressure on water flowing through the hole in the trolley .The pressure on water flowing through the hole will produce a velocity(1) in it perpendicular the direction of momentum of the trolley(2) , and thus the water will move in the direction of the vector sum of 1 and 2 . After this is your question of kinetic energy ; every particle in the trolley system , will keep its kinetic energy with it , except some of this kinetic energy would be lost at the hole where from the water flows out. However in an isolated system the water particles would never come out of the hole because of newtons first law
A: Looking at your problem we can see that:
1) We are ignoring friction.
2) For a clearer picture, assume the water can drop a very long distance before it ever hits Earth and that there is no wind resistance.
Momentum must be conserved in the $x-direction$. There are no net forces acting on anything in the $x-direction$. We have assumed that friction is not an issue.
Any water that has fallen out of the trolley retains the $x-velocity$ of the trolley. After all, there is nothing to resist its forward motion. Every single part of our system retains its original $x-velocity$. The momentum in the x-direction remains the same because all of the masses making up the whole system proceed at a constant velocity in the $x-direction$.
The kinetic energy of the entire system will increase because gravity will perform work, $W=F*D$, on some of the water in the vertical direction.
To completely characterize this system mathematically is no easy task...but it would definitely be fun! The complete solution to this problem involves calculus, Newton's Laws of Motion, and hydraulics with an approximation of Bernoulli's Equation. The water jet velocity approximation requires Bernoulli's Equation.
A: Think of it differently.  Rather than a single train car with a tank of water, you have a train of many individual cars (which can, if you wish, contain water).  If once a second you disconnect one car from the end of the train, what happens to the momentum?
