pipe to know how the inner diameter changes. Rank in order from largest to smallest the gas speeds va, vb, and vc at points a, b, and c. Explain in complete sentences.

To determine the gas speeds (va, vb, and vc) at points a, b, and c in a pipe where the inner diameter changes, we need to consider the principle of continuity and Bernoulli's equation.

1. Start by applying the principle of continuity, which states that the mass flow rate is constant in an incompressible fluid flowing through a pipe. Mathematically, this is expressed as A1v1 = A2v2 = A3v3, where A represents the cross-sectional area and v represents the velocity.

2. Since the inner diameter changes, the cross-sectional areas (A) at points a, b, and c will be different. Assume that the inner diameter at point a (Da) is the largest, followed by point b (Db), and then point c (Dc).

3. According to Bernoulli's equation, the total energy of a fluid flowing through a pipe remains constant between any two points. It states that P + 1/2ρv^2 + ρgh = constant, where P represents pressure, ρ is the density, v is the velocity, g is the gravitational constant, and h is the height.

4. Now, let's consider the pressure terms. Bernoulli's equation assumes the fluid is incompressible, so let's disregard compressibility effects for simplicity.

- At point a, since the diameter is largest, the velocity (va) will be the smallest, and therefore the pressure (Pa) will be the largest.

- At point b, where the diameter is smaller than at point a, the velocity (vb) will be larger compared to va due to the principle of continuity. Consequently, the pressure (Pb) will be smaller compared to Pa.

- At point c, where the diameter is the smallest, the velocity (vc) will be the largest due to the principle of continuity. Thus, the pressure (Pc) will be the smallest compared to both Pa and Pb.

5. Based on the above explanations, we can rank the gas speeds va, vb, and vc in descending order as follows: vc > vb > va.

In conclusion, in a pipe where the inner diameter changes, the gas speed is highest at point c (vc), followed by point b (vb), and lowest at point a (va). This order is determined by applying the principle of continuity and understanding Bernoulli's equation.