Can someone explain a simple physics question for me?

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The discussion centers on understanding the use of vector components in mechanical energy calculations. Participants seek clarification on why forces are expressed as rcostheta and rsintheta, suggesting this simplifies vector components into x and y forms. There is confusion regarding the application of these components when calculating total potential energy, particularly when dealing with angles. An example involving a ball fired at 600m/s at a 30-degree angle highlights discrepancies in calculating total mechanical energy. The conversation emphasizes the need for a clearer explanation of these concepts in the context of conservative forces and energy conservation.
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Okay this isn't really a problem, but I want it explained too me instead of the books way. We are doing a section on mechanical energy, and for the entire course with vectors, force, etc, we have always used rcostheta and rsintheto to find an x and y force.

Now tell me why we do this? Is this to elimate the forces from a vector component form to a basic x y form, where y is a 90 degree value, and x is a 0 degree value?

Also another question. We used the equation for change in mechanical energy where energy potential, and kinetic energy where the same regardless of its position along a path when looking at conservative forces.

I used rcostheta and rsintheta to plug in values for velocity when they where at angles, but I noticed that this data does not accurately give the TOTAL potential energy correctly.

To use an example say that

a ball is fired at 600m/s at 30 degrees with a mass of 24kg.

To plug this into the isloated system model for velocity I had to use velocity x sin30

but when it asked for its total mechanical energy at the balls maximum height I used 1/2mvf^2 and just plugged in 600.
 
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