- #1
Zatman
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- 0
Homework Statement
Show that for any trajectory r(t) the acceleration can be written as:
[itex]\mathbf{a}(t)=\frac{dv}{dt}\hat{T}(t)+\frac{v^2}{\rho}\hat{N}(t)[/itex]
where v is the speed, T is a unit vector tangential to r and N is a unit vector perpendicular to T, at time t. rho is the radius of curvature.
2. The attempt at a solution
[itex]\mathbf{a}=\frac{d\mathbf{v}}{dt}=\frac{d}{dt}(v\hat{T})[/itex]
[itex]=\frac{dv}{dt}\hat{T}+v\frac{d\hat{T}}{dt}[/itex]
Since we can write
[itex]\frac{1}{\rho}\hat{N}=\frac{d\hat{T}}{dt}[/itex]
This gives
[itex]\mathbf{a}=\frac{dv}{dt}\hat{T}+\frac{v}{\rho}\hat{N}[/itex]
I can't see where to get the other 'v' from, in the second term.