# Why water coming out from narrow pipe hurt more than same amount of water coming out from broad pipe with same flow rate?

If you take a narrow and a broad pipe and make same amount of water pass through I.e same flow rate .

And stand at front of both of them

Water from Narrow one hurts more!

Why although same amount water of water coming out in unit time?

Does it due to water molecule velocity in narrow pipe is more(as we know if they have same flow rate, if you want diagram please let me know)?

Why doesn't broad one hurts? How high velocity make us feel high pressure?

If the flow rate is a fixed volume $$V$$ per second, the speed of the water is $$u$$ and the cross-sectional area of the pipe is $$A$$ then we have

$$V = Au$$

If the density of the water is $$\rho$$ then a mass $$\rho V$$ of water per second is hitting you at speed $$u$$. Lets make the simplest assumption, which is that all of the momentum of this water is transferred to your body. Then the force exerted by the water on your body is equal to the rate of change of momentum of the water, which is

$$\displaystyle F = \rho V u = \frac {\rho V^2}{A}$$

If we also assume that the jet does not spread out after leaving the pipe, then this force is applied over an area $$A$$, so the pressure exerted by the water on your body is

$$\displaystyle \frac F A = \frac {\rho V^2}{A^2}$$

Since $$\rho$$ and $$V$$ are fixed, the pressure is inversely proportion to the square of the pipe area $$A$$. This is why a water jet from a narrow pipe is more painful than from a wider pipe.

• Do we feel dynamic pressure or total pressure when water coming out of a pipe hits our hand some distance away from exit of the pipe? If it's total pressure, then why do we feel more force from fluid in narrower pipes than broader? If not, why don't we feel static pressure? Commented Mar 27, 2023 at 17:10

The velocity of the water coming out of the pipe depends upon the flow rate and the cross sectional area. If the flow rates are equal, and the cross sectional area is smaller, the velocity must be greater.

That a higher velocity stream "hurts" more is due to our neurological make-up. I suppose with a different make-up, a high velocity stream might feel different from a low velocity stream in some way, without either of them "hurting" at all. But, as we are constituted, we do feel pain from the streams, and more from the higher velocity stream.

• If our "neurological make-up" tells us that trauma A hurts more than trauma B, it's usually because trauma A did more damage. It takes energy to damage your tissues, and the higher-velocity water stream carries more kinetic energy per unit volume than the lower-velocity stream. In other words, there's a real physical explanation for why the high velocity stream "hurts more." Commented Dec 4, 2021 at 16:55
• Do we feel dynamic pressure or total pressure when water coming out of a pipe hits our hand some distance away from exit of the pipe? If it's total pressure, then why do we feel more force from fluid in narrower pipes than broader? If not, why don't we feel static pressure? Commented Mar 27, 2023 at 17:10
• No. If the flow rate is fixed, the larger the aperture that the water flows through, the lower the pressure will be. Commented Mar 27, 2023 at 17:40

Assuming that the fluid is ideal, a narrow pipe decreases the cross sectional area for the water to go through, but we have assumed that we keep the volume flow rate constant, that means that velocity of fluid coming from narrow pipe will have to increase, cf. the continuity equation for ideal fluids. Once we conclude that velocity from narrow is greater than velocity from broader, a simple use of Bernoulli's equation and the relation that p=mv and that force is the time rate of change of momentum, will show that the narrow stream applies more force.

• Do we feel dynamic pressure or total pressure when water coming out of a pipe hits our hand some distance away from exit of the pipe? If it's total pressure, then why do we feel more force from fluid in narrower pipes than broader? If not, why don't we feel static pressure? Commented Mar 27, 2023 at 17:10