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?
Best Answer
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.