Is acceleration magnitude constant for an object moving along an ellipse?

AI Thread Summary
The discussion focuses on determining the magnitude of the acceleration vector for an object moving along an elliptical path defined by the equations x(t)=acos(ωt) and y(t)=bsin(ωt). The acceleration vector is derived from the velocity components, leading to the conclusion that acceleration is proportional to the position vector r(t) and includes a negative factor of ω². Participants clarify that both r and a are vectors, and the magnitude of a vector can be calculated using the square root of the sum of its components squared. The main question remains whether the magnitude of the acceleration vector is constant over time. The conversation emphasizes the need for clarity in vector calculations and the relationship between acceleration and position in elliptical motion.
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Homework Statement


An object is moving along an ellipse which is described by x(t)=acos(\omegat) and y(t)=bsin(\omegat). Determine magnitude of acceleration vector as a function of parameters a, b, and \omega. Is magnitude of acceleration vector constant over time?

Homework Equations


r(t)=x(t)x+y(t)y
v=dx/dtx+dy/dty

The Attempt at a Solution


r(t)=acos(\omegat)x+bsin(\omegat)y
a(t)=dvx/dtx+dvy/dty
v=-a\omegasin(\omegat)x+b\omegacos(\omegat)y
a=-a\omega^{}2cos(\omegat)x-b\omega^{}2sin(\omegat)y=-\omega^{}2(acos(\omegat)x+bsin(\omegat)y
a=-\omega2r(t)
 
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I assume that you mean the unit vectors along the axes x,y by "x" and "y" and both "r" and "a" are vectors. Your last equation is the relation between the vector of acceleration and the position vector. What is the magnitude of acceleration?

ehild
 
Am I approaching it incorrectly? Both "r" and "a" are vectors. I assume I have to find magnitude of vector "r" but am not certain how to do it.
 
Do you know how to calculate the magnitude of a vector from its components?

ehild
 
I believe I calculate it by doing the square root of its components squared. Is that right?
 

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