I The time derivative of kinetic energy

AI Thread Summary
The discussion focuses on the time derivative of kinetic energy, expressed as T(𝑝) = 𝑝²/(2m), and its relationship to force. It establishes that the derivative dT/dt equals 𝑣·𝑭, linking kinetic energy to Newton's second law where 𝑝 = m𝑣 and 𝑭 = d𝑝/dt. The calculations show that the change in kinetic energy can be derived from the momentum and force relationship. This reinforces the connection between energy, momentum, and force in classical mechanics. The thread emphasizes the mathematical derivation and conceptual understanding of these physical principles.
LagrangeEuler
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Lets consider T(\vec{p})=\frac{\vec{p}^2}{2m}=\frac{\vec{p}\cdot \vec{p}}{2m}. Then \frac{dT}{dt}=\vec{v}\cdot \vec{F}.
And if we consider
T=\frac{p^2}{2m} than \frac{dT}{dt}=\frac{1}{2m}2p\frac{dp}{dt}
Could I see from that somehow that this is \vec{v}\cdot \vec{F}?
 
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LagrangeEuler said:
Lets consider T(\vec{p})=\frac{\vec{p}^2}{2m}=\frac{\vec{p}\cdot \vec{p}}{2m}. Then \frac{dT}{dt}=\vec{v}\cdot \vec{F}.
And if we consider
T=\frac{p^2}{2m} than \frac{dT}{dt}=\frac{1}{2m}2p\frac{dp}{dt}
Could I see from that somehow that this is \vec{v}\cdot \vec{F}?
Well, ##\vec {p} = m \vec {v}## and ##\vec F = \frac{d\vec p}{dt}## is Newton's second law.
 
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