An airplane is cruising at an altitude of 10 km . The pressure outside the craft is 0.287 atm; w

step 1 We have a airplane cruising at 10 kilometers inside the airplane. We have a pressure of one atmo

An airplane is cruising at an altitude of 10 km . The...

An airplane is cruising at an altitude of 10 $\mathrm{km}$ . The pressure outside the craft is 0.287 atm; within the passenger compartment the pressure is 1.00 $\mathrm{atm}$ and the temperature is $20^{\circ} \mathrm{C}$ . A small leak occurs in one of the window seals in the passenger compartment. Model the air as an ideal fluid to find the speed of the stream of air flowing through the leak.

Key Concepts

Bernoulli's Equation

Bernoulli's Equation is a principle in fluid dynamics that relates the pressure, velocity, and height along a streamline in an ideal fluid flow. It shows that as fluid moves from a region of high pressure to a region of low pressure, the pressure energy is converted into kinetic energy, allowing one to compute the velocity of the fluid from a known pressure difference.

Pressure Differential

Pressure differential is the driving force behind fluid flow in many physical systems. It is the difference in pressure between two regions, which causes the fluid to accelerate from the high-pressure area to the low-pressure area, making it a crucial concept in problems involving fluid leak or flow through an opening.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas, and it can be used to calculate the density of a gas when pressure and temperature are known. Understanding the gas density is essential when applying Bernoulli's Equation, as the density of the fluid is a key parameter in determining the conversion of pressure energy into kinetic energy.

Ideal Fluid Approximation

The ideal fluid approximation assumes that the fluid is incompressible and has no viscosity. This simplification removes the complexities associated with energy losses due to friction and compressibility and allows for the straightforward application of conservation principles, such as Bernoulli's Equation, in order to estimate characteristics like the speed of fluid flow through an opening.

Transcript

00:01 We have a airplane cruising at 10 kilometers inside the airplane.

00:13 We have a pressure of one atmosphere of 20 degrees celsius.

00:23 And outside, we have a pressure outside of 0 .287 atmospheres.

00:36 And the question is, there was a small hole in one of the wind of the passengers, what would be the speed that this air will be flowing here? so what is this speed? in order to figure this out, we can start out with the bernoulli's equation.

01:01 So we have p1 or the inside pressure, i'll say.

01:07 So p1 is inside.

01:10 So plus row gy1 plus 1 half row v1 squared equals p2 plus row gy2 plus row gy2, it's a bit messy, row gy2 plus one half row v2 squared.

01:35 And all these rows is the pressure, the density of air.

01:42 Is to make some of the assumptions, to make some of the, to cancel out some of these variables, we're going to make some assumptions.

01:53 We're going to assume that our setup is going to look something like this where here.

02:07 The best way to imagine this might be some of the tubes that we saw in our examples.

02:12 So let's say we have a tube that looks like this.

02:19 And we have two different pressures here and here.

02:24 So we have p2 here and p1 here.

02:27 And imagine our the velocity one we're looking at is so the velocity of this part.

02:38 Of this part.

02:43 So imagine this is all air...

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