Video Transcript
A smoothly flowing liquid flows through a pipe of cross-sectional area 𝐴 one at a speed 𝑣 one. The liquid then passes into a pipe of cross-sectional area 𝐴 two smoothly flowing at a speed 𝑣 two. Which of the following formulas correctly represents the ratio of the liquid’s speed in the first pipe to its speed in the second pipe? (A) 𝐴 one plus 𝐴 two all divided by two. (B) 𝐴 one multiplied by 𝐴 two all divided by 𝐴 one minus 𝐴 two. (C) 𝐴 two divided by 𝐴 one. (D) 𝐴 one divided by 𝐴 two. (E) 𝐴 one squared plus 𝐴 two squared all to the power of one-half.
Our problem statement talks about two pipes that are connected. Let’s say that this is our first pipe and this is the second one. Liquid enters the first pipe at a speed 𝑣 one, and it exits the second at a speed 𝑣 two. Along with this, the cross-sectional area of the first pipe is 𝐴 one and that of the second pipe is 𝐴 two. We want to figure out which of these five expressions correctly relates the ratio of the liquid’s speed in the first pipe, that’s 𝑣 one, to its speed in the second pipe, that’s 𝑣 two. In other words, we want to solve for the ratio 𝑣 one divided by 𝑣 two.
To begin doing this, let’s clear some space on screen, and let’s recall something called the continuity equation for fluids. When a fluid is incompressible so that it has a constant density, that equation is written this way. It says that any cross-sectional area through which the fluid flows multiplied by the fluid’s speed at that point is equal to any other cross-sectional area through which the fluid smoothly flows multiplied by its speed there. Note that we’re using this form of the continuity equation because we’re working with a liquid, which we can assume is incompressible.
In our scenario, the first area is the entrance area of our first pipe, and the second area is the exit area of our second pipe. 𝐴 one 𝑣 one equals 𝐴 two 𝑣 two, and we can start to rearrange this equation to get the fraction 𝑣 one divided by 𝑣 two. To do so, we’ll divide both sides of this equation by 𝑣 two times 𝐴 one. This means that on the left-hand side, 𝐴 one cancels from numerator and denominator, and on the right 𝑣 two cancels out. What we find is that 𝑣 one divided by 𝑣 two equals 𝐴 two divided by 𝐴 one. Reviewing our answer options, we see this agrees with option (C). The ratio of the speed of this fluid as it enters the first pipe to the speed of the fluid as it exits the second is 𝐴 two over 𝐴 one. We choose option (C) as our answer.
