Keplers laws and kinetic energy

AI Thread Summary
The discussion focuses on deriving the kinetic energy equation for a planet in an elliptical orbit, specifically Kinetic energy = 1/2 m(ṙ²) + 1/2 m(r²)(θ̇²). Participants clarify that ṙ represents the radial velocity, while rθ̇ indicates tangential velocity, which is a component of linear velocity. The distinction is made that θ̇ is angular velocity, and ṙ/dθ is not a velocity but rather a derivative that lacks a specific name. The conversation emphasizes the importance of understanding these components in the context of orbital mechanics. Overall, the thread provides insights into the relationships between different types of velocities in orbital dynamics.
sheetman
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Homework Statement



As part of my coursework I've had to show that kinetic energy can be written in a certain way. As a cheat I used this as a starting point and worked my way back to the following equation

Kinetic energy = \frac{1}{2}m\dot{r^{2}} + \frac{1}{2}mr^{2}\dot{\theta}^{2}

r dot is the derivative of radius with respect to theta, and theta dot is the derivative of theta with respect to time.

Homework Equations



(edit: This is for the kinetic energy of a planet in an elliptical orbit around the sun)

The Attempt at a Solution



the first part of the equation for kinetic energy is angular velocity, but I can't for the life of me figure out what r\frac{d\theta}{dt} is. Is it another way of calculating angular velocity or have I probably messed up the algebra somewhere?

Thanks
 
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You may recognize it as ωr since ω=dθ/dt. The other way you can view that term is as 1/2 (mr22.
 
So the second term is angular velocity, so what kind of velocity is dr/d(theta)?
 
sheetman said:
So the second term is angular velocity
Not exactly. First, the second term is an energy, not a velocity. Second, angular velocity is dθ/dt while r dθ/dt is the tangential velocity, a component of the linear velocity. Depending on how you choose to interpret the second term, you can say it's a function of either the angular velocity or the tangential velocity.
so what kind of velocity is dr/d(theta)?
It's the component of velocity perpendicular to the tangential velocity. It's the radial velocity of the body.
 
Hero, thanks very much!
 
I just noticed you said dr/dθ, not dr/dt. It's dr/dt which is the radial velocity. I don't know of a name for dr/dθ. It's not a velocity, though.
 
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