Solving Streamlines for V = (ay)i + bj

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In summary, the given flow is described by the velocity field V = (yi + 2j) m/s, where coordinates are measured in meters. The equation for the streamline passing through point (6,6) is (y^2)/2 = 2x+6. To find the coordinates of the particle that passed through the point (-3,0) 2 seconds earlier at t = 1 seconds, the use of Lagrangian mechanics is required.
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banerjeerupak
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Homework Statement



A flow is described by velocity field V=(ay)i + bj, where a = 1, and b = 2. Co-ordinates are measured in meters. Obtain the equaion for the streamline passing through point (6,6). At t = 1 seconds, what are the co-ordinates of the particle that passed through the point (-3,0) 2 seconds earlier.

Homework Equations



V = (ay) i + bj

The Attempt at a Solution



From the values of a and b, we have V = yi + 2j.
dy/dx = v/u = 2/y.

Integrating throughout,
(y^2)/2 = 2x + c.

@(6,6) -> 18 = 12 + C

C = 6

therefore, (y^2)/2 = 2x+6

I can't figure out how to fine the other part of the problem. i.e. "At t = 1 seconds, what are the co-ordinates of the particle that passed through the point (-3,0) 2 seconds earlier. ".
 
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I'm assuming that this requires the use of Lagrangian mechanics, which I haven't learned yet. Any help is greatly appreciated.
 

FAQ: Solving Streamlines for V = (ay)i + bj

What is the meaning of V = (ay)i + bj?

The equation V = (ay)i + bj represents a velocity vector in a two-dimensional space, where ay is the velocity in the x-direction and bj is the velocity in the y-direction. This equation is used to describe the flow of a fluid or gas in a streamline.

How do you solve for V = (ay)i + bj?

To solve for V = (ay)i + bj, you can use the principles of vector addition and trigonometry. First, break down the vector into its x and y components. Then, use the Pythagorean theorem to find the magnitude of the vector. Finally, use inverse trigonometric functions to find the direction of the vector.

What is the significance of ay and bj in the equation V = (ay)i + bj?

The values of ay and bj represent the velocity in the x and y directions, respectively. These values are important in determining the direction and magnitude of the fluid or gas flow in a streamline.

How does solving for V = (ay)i + bj help in studying fluid dynamics?

Solving for V = (ay)i + bj allows scientists to understand the direction and speed of fluid flow, which is crucial in studying fluid dynamics. This equation is often used to model and predict the behavior of fluids in various systems, such as in aerodynamics and hydrodynamics.

Can V = (ay)i + bj be used to solve real-world problems?

Yes, V = (ay)i + bj can be used to solve real-world problems in fluid dynamics. This equation is commonly used in engineering, meteorology, and other fields to analyze and predict fluid flow behavior in various systems and processes.

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