Laws of motion Questions
1. Show that for a single particle with constant mass the
equation of motion can be put in the form $\frac{dT}{dt} = \vec{F} ⋅\vec{v}$
where $T$ is kinetic energy, $\vec{F}$ is the force applied and $\vec{v}$ is velocity.
From Newton's second law of motion, we have,
$\frac{d \vec{P}}{dt} = \vec{F} \\$
$m⋅\frac{d \vec{v}}{dt} = \vec{F} \\$
$m⋅\frac{d \vec{v}}{dt} ⋅ \vec{v}= \vec{F} ⋅ \vec{v}\\$
$m⋅\frac{d }{dt} (\frac{v^2}{2})= \vec{F} ⋅ \vec{v}\\$
$\frac{d }{dt} (\frac{mv^2}{2})= \vec{F} ⋅ \vec{v}\\$
$\implies \frac{dT}{dt} = \vec{F} ⋅\vec{v}$