Solving Parabolic Coordinates Homework: Find x,y; Show Kinetic Energy

In summary, the conversation discusses finding the values of x and y in terms of ξ and η and using them to calculate the kinetic energy of a particle of mass m. The solution is found by solving for x and y in terms of ξ and η, and then plugging them into the equation for kinetic energy. After some trial and error, the correct values for x and y are found and the final equation for kinetic energy is derived.
  • #1
metgt4
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Homework Statement


(ξ,η) in a plane are defined by η = (x2 + y2)1/2 - x and ξ = (x2 + y2)1/2 + x

Find x and y in terms of ξ and η. Show that the kinetic energy of a particle of mass m is:

T = (m/8)(ξ + η)(ξ'2/ξ + η'2/η)


The Attempt at a Solution



My attempt is scanned and appended in this post. I found x and y in terms of η and ξ, but am not sure where to go from there in order to find the kinetic energy in terms of the two. If somebody could point me in the right direction, I'd really appreciate it.

Thanks!
Andrew
 

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  • #2
Nevermind, got it figured out. I went wrong in finding x and y as well.
 

FAQ: Solving Parabolic Coordinates Homework: Find x,y; Show Kinetic Energy

What are parabolic coordinates?

Parabolic coordinates are a type of coordinate system used in mathematics and physics to describe points in three-dimensional space. They are based on the shape of a parabola and are often used in problems involving objects moving along a parabolic path.

How do you solve for x and y in parabolic coordinates?

To solve for x and y in parabolic coordinates, you can use the equations x = ρ cos(θ) and y = ρ sin(θ), where ρ is the distance from the origin to the point and θ is the angle between the point and the positive x-axis. These equations can be derived from the definition of parabolic coordinates.

What is the relationship between parabolic coordinates and kinetic energy?

The relationship between parabolic coordinates and kinetic energy is that parabolic coordinates can be used to describe the motion of an object in a parabolic path, which is a common type of motion in physics. Kinetic energy is the energy an object possesses due to its motion, so it can be calculated using the equations of motion in parabolic coordinates.

How do you show kinetic energy in parabolic coordinates?

To show kinetic energy in parabolic coordinates, you can use the formula KE = 1/2*m*(dx/dt)^2, where m is the mass of the object and (dx/dt) is the derivative of the position with respect to time. This formula can be adapted to parabolic coordinates by using the appropriate equations for x and y in terms of ρ and θ.

Can you provide an example of solving a problem using parabolic coordinates and kinetic energy?

Sure, for example, if you have an object moving along a parabolic path with a mass of 2 kg and a speed of 5 m/s at a distance of 3 meters from the origin, you can use the equations x = ρ cos(θ) and y = ρ sin(θ) to find the values of x and y in parabolic coordinates. Then, you can use the formula KE = 1/2*m*(dx/dt)^2 to calculate the kinetic energy at that point. This can be a useful tool for analyzing the motion of objects in parabolic paths in physics problems.

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