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A horizontal pipe carries water in the p...

A horizontal pipe carries water in the positive direction of x. Small identical holes are made at three different places along the pipe such that streams of water from them come out vertically upwards. The height upto which the water stream rise from the three holes

A

increase in the +Ve x-direction

B

decreases in the +ve x-direction

C

are the same in all

D

are the same for the first two and less for the third hole

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To solve the problem of determining the height to which water streams rise from three holes in a horizontal pipe, we will use Bernoulli's theorem. Here’s a step-by-step breakdown of the solution: ### Step 1: Understand the Setup We have a horizontal pipe carrying water, and there are three holes made at different points along the pipe. Water flows out of these holes vertically upwards. ### Step 2: Apply Bernoulli's Theorem According to Bernoulli's theorem for a fluid flowing in a horizontal direction, the equation can be expressed as: \[ P + \frac{1}{2} \rho v^2 = \text{constant} \] where: - \( P \) is the pressure, - \( \rho \) is the density of the fluid (water in this case), - \( v \) is the velocity of the fluid. ### Step 3: Analyze the Flow Since the pipe is horizontal, the gravitational potential energy term does not come into play. The pressure and kinetic energy terms are the only ones we need to consider. ### Step 4: Consider the Effect of Holes When water exits the holes, it will rise to a certain height due to the pressure at that point. The height \( h \) to which the water rises can be related to the pressure and velocity using the principle of conservation of energy. ### Step 5: Determine Pressure and Velocity at Holes If the velocity of the water is the same at all three holes (which is true since the cross-sectional area of the pipe does not change), then the pressure at all three holes will also be the same. This is because, according to Bernoulli’s principle, if the velocity remains constant, the pressure must also remain constant. ### Step 6: Conclude the Height of Water Rise Since the pressure is the same at all three holes and the velocity of the water is also the same, the height to which the water rises from each hole will be identical. Thus, the height \( h \) will be the same for all three holes. ### Final Answer The height to which the water streams rise from the three holes is the same.
To solve the problem of determining the height to which water streams rise from three holes in a horizontal pipe, we will use Bernoulli's theorem. Here’s a step-by-step breakdown of the solution: ### Step 1: Understand the Setup We have a horizontal pipe carrying water, and there are three holes made at different points along the pipe. Water flows out of these holes vertically upwards. ### Step 2: Apply Bernoulli's Theorem According to Bernoulli's theorem for a fluid flowing in a horizontal direction, the equation can be expressed as: \[ ...
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