Speed of body traveling in elliptical path

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A body traveling in an elliptical path is described by the equation r = sd/(1-s*cosP), with constant angular speed. To find the body's speed, the magnitude of the velocity vector must be calculated using the equation v = dr/dT*rhat + r*dP/dT*Phat. The key is to evaluate dr/dt by applying the chain rule, recognizing that P is a function of time. The solution involves differentiating r with respect to time, incorporating the angular speed. Ultimately, the correct expression for speed is v = rw[1+{(r*sinP)/d}^2]^(1/2).
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Homework Statement


A body follows an elliptical path defined by r = sd/(1-s*cosP), where s and d are constant. If the angular speed is constant (dP/dT = w), show that the body's speed is v = rw[1+{(r*sinP)/d}^2]^(1/2)


Homework Equations


v = dr/dT*rhat + r*dP/dT*Phat; P = theta, I am not sure how to insert a theta symbol


The Attempt at a Solution


Right now I am thinking that I should find dr/dt. But as I see it since, there is no time dependence in the original equation, dr/dt = 0. Am I safe to take the derivative of r and use it as v? I am thinking no, since v is the magnitude of the velocity vector. Thanks for the help!
 
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Varnson said:

Homework Statement


A body follows an elliptical path defined by r = sd/(1-s*cosP), where s and d are constant. If the angular speed is constant (dP/dT = w), show that the body's speed is v = rw[1+{(r*sinP)/d}^2]^(1/2)

Homework Equations


v = dr/dT*rhat + r*dP/dT*Phat; P = theta, I am not sure how to insert a theta symbol

The Attempt at a Solution


Right now I am thinking that I should find dr/dt. But as I see it since, there is no time dependence in the original equation, dr/dt = 0. Am I safe to take the derivative of r and use it as v? I am thinking no, since v is the magnitude of the velocity vector. Thanks for the help!
You've got the parts of it right, but you're a little confused.

\vec{v} = \frac{d}{dt} (r \hat{r} ) = r \omega \hat{\theta} + \hat{r} \frac{dr}{dt}

From this, you can write down the magnitude of the velocity, | \vec{v} |

The only missing piece, is to evaluate dr/dt. This you can do from the elliptic equation, with the time dependence embedded in \theta(t).
 
Use the chain rule. d/dT=d/dP*dP/dT. P is a function of time and so is r through it's dependence on P.
 
I figured it out, I was stuck after I took the derivative, then i solved for r/d and the answer was right there in front of me! Thanks for the help!
 
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