Speed of body traveling in elliptical path

[Varnson]

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!

 

[Gokul43201]

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).

 

[Dick]

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.

 

[Varnson]

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!