In which bucket , should the water fill it up first?

Question

Here , we have two water filters and buckets .

Now , we know that as the water falls down , it’s speed increases because of acceleration due to gravity.

Let us say I started both the filters and once the water reaches the bottom of both the buckets.of course , water from 2nd filter would have had reached first at the surface of 2nd bucket.

Assume that exactly when the water touches the bucket form both the filters. Only then , I start the timer. Now , I am recording or noting the time it is taken by both filters to fill the bucket.

Volume is same of both buckets.

The Q is : Which bucket will fill up first ?

I think both the buckets would fill up at the same time.

Now , I also noticed as the water keeps on dripping at longer heights. There is a separation between water molecules present. You can see most of the water at below the tap and it’s amount kind of like looks fo reduce as you go down.

This is what confused me. From this observation , I think 2nd bucket should fill up faster than the 1st one.

Why ? Because if I imagine the separation between the bucket and water at an extreme level. Then , I can see like drops filling up the bucket and not like a stream of flowing water.

Is this correct ? Also , I’m not getting how should I write an equation for this.

I did research about it. It says that :

A decrease in temperature caused the water molecules to lose energy and slow down, which results in water molecules that are closer together and a decrease in water volume. When water is heated, it expands, or increases in volume. When water increases in volume, it becomes less dense.

Therefore , can I say my observation is true only depending on the temperature?

Answer 1

IMHO (note its an opinion) in both cases the cup will fill at the same time.

The reason is that the flow rate is controlled by the top vessel. Let's say it is $\dot m$.

However, $$\dot m = v\cdot A$$

where:

• A is the cross-section
• v is the velocity

In a unit of time, a packet of water $\Delta m$ released in time $\Delta t$ is stretched.

However, the bottom and top should arrive at the same time.

figure 1: pouring oil (source: vectorstock)

The basic idea is that:

• every $dt$ a packet of fluid is released (assuming it starts at $t_0$)
• the first edge of the package will reach the cup after $t_f$.
• The first edge of the next packet will reach the cup after $t_f + dt$.

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consider splashing

if you take into account the drop that are spilled from the cup because of the high speed then its no contest, the 2nd cup fills quicker..

Answer 2

A similar question is asked about conservation of mass in water from a faucet here and is partially answered in the comments.

The buckets should fill up at the same time, since the mass flowrate at any point in the falling stream is equal. As velocity increases further down the stream, the cross-sectional area of the stream will decrease so that $\rho ua$ remains constant. If mass flowrate is constant, then the mass flow entering both buckets will be the same regardless of their height.

The smaller stream you were noticing at the bottom is because of conserved mass, higher velocity means smaller area. If we begin to factor in physical effects such as droplet forming as the stream separates, I'm not sure about the answer.

The point about the temperature changing density is incorrect:

A decrease in temperature caused the water molecules to lose energy and slow down, which results in water molecules that are closer together and a decrease in water volume. When water is heated, it expands, or increases in volume. When water increases in volume, it becomes less dense.

Liquids, including water, are generally treated as incompressible, meaning constant density. Liquid water does not change its density as a function of temperature, except for phase changes (freezing, melting, evaporating). Now, viscosity (resistance to flow) of a fluid (gas or liquid) is temperature dependent, so colder water will have higher viscosity and thus more energy lost to drag forces. But I would say considering viscosity would fall under the umbrella of "other" effects such as surface tension, droplet formation, etc. which is beyond the conservation analysis I discussed.

Answer 3

The top bucket will fill up first because the water needs less time to fall from the reservoir to the bucket.