Can someone explain a simple physics question for me?

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The discussion centers on the application of vector components in mechanical energy calculations, specifically using the equations rcostheta and rsintheta to resolve forces into x and y components. The user questions the rationale behind this method, suggesting it simplifies the representation of forces by aligning them with standard Cartesian coordinates. Additionally, the user highlights a discrepancy in calculating total mechanical energy when applying these components to a scenario involving a ball fired at 600 m/s at a 30-degree angle, indicating that the potential energy calculations may not align with expected outcomes in conservative force systems.

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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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