## the unit tangent/normal vectors to motion+their derivatives

1. The problem statement, all variables and given/known data

2. Relevant equations
magnitude of n, u=1, u is in the direction of the velocity u=V/V

3. The attempt at a solution
The 1st part is easy, I wrote:

But I can't do the second part, I read about 10 different sources about these unit vectors, but now I'm even more confused. Especially about so called "radial" and "transverse" unit vectors, they are not "tangential" and "normal" unit vectors?

Anyway, for this part I tried analytical approach:

But not sure what to do next. Or another approach:

Any help will be appreciated, I spent about 2 days on this problem alone
 Mentor Use the chain rule to express $\math d \hat u / dt$ in terms of $\math d \hat u / ds$.
 Your alternate approach using theta will work. Let $$\rho\mbox{d\theta}=ds$$ Integrate with respect time. V(s) = ds/dt and $$\rho\frac{d\theta}{dt}=\rho\omega$$ and $$\omega{ds}= \mbox{magnitude of V}$$

## the unit tangent/normal vectors to motion+their derivatives

ok, but I don't know what to do next :(

$$\frac{d \hat u}{ds}=\frac{d \hat u}{ds}$$$$\frac{ds}{dt}$$

 Mentor You know $$\frac{d\hat u}{ds}$$ from part (a), and you should know $$\frac{ds}{dt}$$.