The Shape of Falling Water
If you have ever looked at water being poured or coming out of a tap you will probably have noticed the shape. The stream of water is wider at the top than the bottom. If you look carefully then you see the shape is a curve. The decrease is much more pronounced at the top than the bottom.
The reason for the shrinking width is fairly obvious. The water is falling under gravity so like everything that falls, it accelerates. This means that the water at the top is moving more slowly than the water at the bottom. We also know that the amount of water flowing stays constant. Virtually all of the water that leaves the tap reaches the sink below.
If the water is moving faster then for the same cross sectional area, more water will be flowing. Therefore because the flow stays constant the cross sectional area must decrease so the hence the width decreases.
Now all that is left is to explain why the shape is a curve rather than a cone shape. There are in fact two separate reasons for this.
Firstly the cross sectional area is inversely proportional to the velocity to maintain a constant flow. What you see, when you look at the flowing water from the side, is the width of the flow. This means that the velocity is inversely proportional to the square of the width. This will cause a curve.
The second reason is because the velocity increase linearly with time not distance. So the increase in velocity over 1 second will be the same at any point. However further down the water is moving faster so over that second the water have moved further. So over 1 cm the velocity will increase by a smaller amount at the bottom than the top.