# Newton's second law and moving through a fluid

It is harder to move an object through a dense fluid like water than compared to with a less dense fluid like air. Is an explanation for this possible with Newton's 2nd law? There are people that say the phenomenon occurs becasue of fluid pressure of fluid density or number of molecules needed to be pushed out of the way when walking. But I was thinking that Newton's 2nd law could explain this phenomenon. My thinking is below.

$$F=ma$$. So heavier particles have less acceleration for a given constant force, which means a heavy particle does not get of the way as faster as a light particle when, say, someone is walking in water. Thus denser fluids are harder to move in because of Newtons 2nd law. Is this correct?

• sure the opposing force is going to be proportional to the density of liquid remember this might change in different fields, gravitational potential being linear near earth your explaination is correct Sep 7, 2021 at 5:52
• Your intuition is correct, the only minor correction is that it's the mass of the fluid in front of the object (density times frontal area) rather than the mass of an individual particle that matters.
– Rick
Sep 7, 2021 at 18:30

The words you're looking for are "viscosity" and "drag"

Viscosity is a physical measure of how resistant a fluid is to deformation. This animation from Wikipedia should make it more apparent. The viscous force is given by: $$F_v = \mu \frac{dv}{dy}$$ where $$\mu$$ is the coefficient of viscosity. This term is primarily dependent on the density of the fluid and any intermolecular forces between it's constituents (along with the temperature.)

Your analogy of pushing heavier molecules is actually incorrect. Water is primarily composed of $$\require{mhchem}\ce{H2O}$$ which has a molar mass of 18 g/mol, but air is (primarily) composed of $$\ce{N2}$$ (28 g/mol), $$\ce{O2}$$ (32 g/mol) and $$\ce{CO2}$$ (44 g/mol) which are are each heavier than water.

Common intuition however points out that it is easier to tread in air than in water, so this argument is flawed.

We instead make use of the fact that air is far less dense than water for a given volume. Water molecules also exhibit hydrogen bonding, a significant intermolecular attractive force. In accordance with the equation, we find that water exerts a much larger viscous force than air because of it's higher coefficient of viscosity

In accordance with Newton's Second Law, the resulting acceleration reduces, making it "harder" to move objects using the same constant force. $$F - F_v = ma_x$$

This assumption is only justified when the flow is laminar. For fluids with higher values of the Reynold Number, this viscous force becomes less dominant, because the flow becomes turbulent (Thanks @Rick for pointing this out)

The force of drag is a model of resistive force for an object moving through a fluid. It is given by $$R = \frac{1}{2}D\rho Av^2$$ where $$D$$ is the drag coefficient, and A is the cross sectional area of the moving object measured in a plane perpendicular to its velocity.

This resistive force, clearly increases with the square of the velocity, and is bound to play a dominating role when you push through a fluid (like in your example). Once again, you can use Newton's second law to show how this makes it "harder" to move through: $$F - R = m. \frac{dv_x}{dt}$$

For reference, the coefficient of viscosity for water is $$8.90 × 10^{-4}$$ Pa.s and for air is $$18.1 × 10^{-6}$$ Pa.s

Hope this helps.

• Viscous drag and pressure drag are two different things and which one dominates is related to the Reynolds number. The OP was correctly describing pressure drag $F=\frac12\,C_D\,\rho\,A\,V^2$ which is independent of viscosity. You're talking about viscous or frictional drag which is only dominant with low Reynolds numbers or very streamlined bodies.
– Rick
Sep 7, 2021 at 17:48
• Also viscosity is largely independent of density. In fact, most gases when compressed to higher densities retain the same viscosity. Oil, which is slightly less dense than water can be much much more viscous.
– Rick
Sep 7, 2021 at 17:50
• To estimate the Reynolds number of walking through water: $Re=\frac{\rho\,V\,D}{\mu} \approx \frac{1 \frac{gm}{{cm}^3} \cdot 5\,m.p.h. \cdot 5 in.}{ 1 \, mP\,s} \approx 300,000$ so well into the turbulent range where the viscous portion of the drag is negligible.
– Rick
Sep 7, 2021 at 18:12
• @Rick, Thank you for pointing that out, I have made changes to my post, and included content based on pressure-related drag resistive force. Sep 7, 2021 at 18:27

At the first glance, your explanation seems to be right. But there is a missing point. Let's see.

Our concentration is about walking through water vs. walking through the air. As you say, the explanation of those peaple who say that "the phenomenon occurs becasue of fluid pressure of fluid density or number of molecules needed to be pushed out of the way when walking" sounds good. The deciding factor here is density. It is a measure of number of particles per unit volume. Forgetting the walking, let us imagine we are trying to push water and air. For the ease of imagination, let's condense 1 m3 of water and air (seperately) to two boxes. Then the air box will weigh 1.29 kg and the box of water will weigh 1000 kg. Obviously it is hard to push the box of water while you can move the box of air with one finger (or five : ) ). The reason for this is that the water contains more molecules than the air within the same amount of volume. Thus this is due to density. Overall, this is explained with the aid of Newton's second law, since we use it when determining the ease of pushing 1.29 kg vs 1000 kg.

Then let's move on to your explanation. Your explanation is based on particles. If I have not misinterpreted your words, paticles and density means two different things. Higher density doesn't imply heavier particles. For instance, 6.022×1023 particles of water (more precisely, one mole) weigh 18.01 g, while the same number of air molecules weigh 28.97 g. Thus it is harder to push an air molecule than a water molecule. Hence, according to your explanation, it will be harder to move through the air than the water!

Thus, in this situation, you should consider the role of density and the mass of one particle as well.

Hope this helps.