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I am exploring the motion of a particle who's path P(x,y) is given by two parametric equations:

x = f(t) = a cos(bt) + cd (1)

y = g(t) = a sin(bt) (2)

To get a tangential velocity for this particle, I differentiated (1) and (2) then combined them using the form sqrt( m

f'(t) = -ab sin(bt) + cd (3)

g'(t) = ab cos(bt) (4)

Tv(t) = sqrt ( (-ab sin(bt) + cd )

Next, I wanted to examine the tangential acceleration of the particle, so I differentiated (5):

Ta(t) = -a

The results of everything so far seem to be as expected. Now I wanted to isolate the horizontal and vertical components of the tangential velocity and acceleration. So I use the following to isolate the horizontal velocity:

Xv(t) = -sin(bt) sqrt( (-ab sin(bt) + cd )

Then differentiated that to get a horizontal acceleration function. This is where things started diverging from the expected results.

Was my choice for the horizontal acceleration function wrong? Should have just multiplied the tangential acceleration function by same term that I used to multiply the tangential velocity by to get a horizontal acceleration function?

Things got even worse when I applied this strategy to the vertical component.

***

It just occurred to me that what I want is a vector for the tangential acceleration, then isolate the x and y components from that.

x = f(t) = a cos(bt) + cd (1)

y = g(t) = a sin(bt) (2)

To get a tangential velocity for this particle, I differentiated (1) and (2) then combined them using the form sqrt( m

^{2}+ n^{2}):f'(t) = -ab sin(bt) + cd (3)

g'(t) = ab cos(bt) (4)

Tv(t) = sqrt ( (-ab sin(bt) + cd )

^{2}+ ( ab cos(bt) )^{2}) (5)Next, I wanted to examine the tangential acceleration of the particle, so I differentiated (5):

Ta(t) = -a

^{2}b^{2}c d cos(bt) / sqrt ( (-ab sin(bt) + cd )^{2}+ ( ab cos(bt) )^{2}) (6)The results of everything so far seem to be as expected. Now I wanted to isolate the horizontal and vertical components of the tangential velocity and acceleration. So I use the following to isolate the horizontal velocity:

Xv(t) = -sin(bt) sqrt( (-ab sin(bt) + cd )

^{2}+ ( ab cos(bt) )^{2}) (7)Then differentiated that to get a horizontal acceleration function. This is where things started diverging from the expected results.

Was my choice for the horizontal acceleration function wrong? Should have just multiplied the tangential acceleration function by same term that I used to multiply the tangential velocity by to get a horizontal acceleration function?

Things got even worse when I applied this strategy to the vertical component.

***

It just occurred to me that what I want is a vector for the tangential acceleration, then isolate the x and y components from that.

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