The water level in the container may be computed by dividing the volume of water in the container by its cross-sectional area. You would probably arrive at the eqaution
h = Qt/A
where t is the time. Q , the flow rate of water and A, the cross-sectional area. Since Q and A are the same for each container, then the water levels will also be the same right? Nah.
The problem with the equation above is that Qt is not really the volume of water in the container at time t. Qt actually represents the volume of water that has left the faucet at time t, not all of which will have made its way into the container. In fact, at an arbitrary time, there will be both water in the container and water between the faucet and the container. Thus, even though the two faucets are emitting the same amount of water every second, it is possible to have more water in one container if it has less water between the faucet and the container.
Let us neglect the pull of gravity to simplify our analysis. Let A be the cross-sectional area of the container to be filled, and a be the cross sectional area of the faucet. Then if the flow rate is Q, using the continuity equation, then the smaller the cross sectional area of the faucet, the faster the water will be traveling. (v = Q/a). If the faucet is at a height H above the bottom of the container, then it will take a time ts =H/v = Ha/Q before the container even begins to fill with water. Therefore, the smaller the faucet area, the less time it takes to start filling the container. In addition to this, the volume of water between the faucet and the container will be less if the faucet area is smaller. This means that more volume of water goes into filling the container, resulting in a higher water level. So, the container placed under the faucet with smaller area will fill up first!
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