- #1
JD_PM
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I don't get why ##F \cdot dr = \frac{mv^2}{2}##
I know this has to be really easy but don't see it.
Thanks.
Particle moving in conservative force field
- Thread starter JD_PM
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Homework Help Overview
The discussion revolves around the relationship between force and kinetic energy in the context of a particle moving in a conservative force field. Participants are exploring the equation ##F \cdot dr = \frac{mv^2}{2}## and its implications.
Discussion Character
- Conceptual clarification, Mathematical reasoning
Approaches and Questions Raised
- Participants are questioning the derivation and meaning of the equation relating force and kinetic energy. There are discussions about the vector nature of velocity and its magnitude, as well as attempts to clarify the mathematical relationships involved.
Discussion Status
The discussion is ongoing, with participants providing insights into the mathematical derivation of force and its relation to velocity. Some participants are exploring different interpretations of the equations presented, while others are seeking clarification on specific terms and definitions.
Contextual Notes
There are references to a textbook that may provide additional context, specifically "Vector Analysis; Schaum's outlines." Participants are also grappling with the definitions and assumptions underlying the equations discussed.
I don't get why ##F \cdot dr = \frac{mv^2}{2}##
I know this has to be really easy but don't see it.
Thanks.
- #2
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What is the derivative of ##\vec v^2## with respect to time?
- #3
JD_PM
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I see what you mean but that is the vector and not the magnitude
- #4
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No it is not. ##\vec v^2 = \vec v \cdot \vec v = v^2##.JD_PM said:
- #5
archaic
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Could you tell me the book's name?JD_PM said:
- #6
JD_PM
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archaic said:
Sure
Vector Analysis; Schaum's outlines
- #7
ehild
Science Advisor
Homework Helper
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I think the author intended to use
## F= \frac{m}{2}\frac{d(v^2)}{dr}##.
It comes from the definition of force
##F=m\frac{d^2r}{dt^2}=m\frac{dv}{dt}##
as ##v=\frac{dr}{dt}##.
Applying chain rule when using derivative with respect to r instead of t:
##F=m\frac{d v}{d r} \frac{d r}{d t}=m v \frac{d v}{d r}=\frac{m}{2}\frac{d(v^2)}{dr}## .
## F= \frac{m}{2}\frac{d(v^2)}{dr}##.
It comes from the definition of force
##F=m\frac{d^2r}{dt^2}=m\frac{dv}{dt}##
as ##v=\frac{dr}{dt}##.
Applying chain rule when using derivative with respect to r instead of t:
##F=m\frac{d v}{d r} \frac{d r}{d t}=m v \frac{d v}{d r}=\frac{m}{2}\frac{d(v^2)}{dr}## .
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