Kinematics of particles: a sample problem with spherical coordinates

[Ellinor]

Homework Statement

Resolve the velocity of the aircraft P into spherical coordinate components 60 seconds after takeoff and find R dot, θ dot and 𝜙 dot for that instant.

Relevant Equations

Hello. I am stuck on a sample problem in my textbook. The sample problem consists of 2 small problems, a and b. More specifically, I do not understand the explanation given in the solution for problem b.

I tried to mark the part that I do not understand with a question mark in the images.
I do not understand why R dot equals 99,2 cos (13,19°) + 30,4 sin(13,19°).

99,2 is the speed of the vr component of v in cylindrical coordinates and 30,4 is the speed of the vz component in cylindrical coordinates, calculated in problem a. How can they calculate the vR component of V in spherical coordinates from that information?

Neither do I understand how they calculate v_theta and v__phi.

I have refered to the image given but it makes me none the wiser. I published my question in engineering since it is an engineering mechanics textbook. Thank you in advance.

 

[Gavran]

$\begin{align} &R=\sqrt{r^2+z^2}\nonumber\\ &\dot R=\frac{\partial R}{\partial r}\dot r+\frac{\partial R}{\partial z}\dot z\nonumber\\ \end{align}$

$\begin{align} &\theta_{sph.}=\theta_{cyl.}\nonumber\\ &\dot\theta_{sph.}=\dot\theta_{cyl.}\nonumber\\ \end{align}$

$\begin{align} &\phi=\arctan\frac zr\nonumber\\ &\dot\phi=\frac{\partial\phi}{\partial r}\dot r+\frac{\partial\phi}{\partial z}\dot z\nonumber\\ \end{align}$

I hope the above equations will be helpful.

 

[Lnewqban]

Welcome, @Ellinor !

Can you geometrically visualize the projection of vectors on the right side of figure (c)?