Is the Angle Between Velocity and Acceleration Constant for a Spiraling Bee?

[negation]

Homework Statement

A bee goes out from its hive in a spiral path given in plane polar coordinates by r = bekt q = ct where b, k, and c are positive constants. Show that the angle between the velocity vector and the acceleration vector remains constant as the bee moves outward.
Solution
r(t) = bekt q(t) = ct

The Attempt at a Solution

r'(t) = bke^(kt)
r' ' (t) = bk^(2)e^(kt)

I'm lost as to what should I be doing with the information to make the jump to the proof.

 

[BvU]

Not good to throw up your hands under 3). See guidelines.

How does one go about calculating an angle between vectors ?

 

[BvU]

He, hello Negation!

Also some more explanation under 1) is needed. You should know better by now! I can deduce some things, so I can guess that $r = b \, e^{kt}$ and $\theta = q\, t$ but this way you make it too difficult for others. At least learn a bit about Go Advanced. It is a safe, easy environment. $\TeX$ is much more powerful, but also a lot tougher. Still worth the investment...

 

[BvU]

Oh, and: if a vector in plane polar coordinates has two components, how come you only write down $\dot r$ and $\ddot r$ (this is shorthand for $dr\over dt$ and $d^2 r \over dt^2$) if you really need $\dot {\vec r}$ and $\ddot {\vec r}$ to say something about the angle ?

 

[negation]

Not good to throw up your hands under 3). See guidelines.

How does one go about calculating an angle between vectors ?

By using the dot product and which by definition states:

A.B = |A||B|cosΘ
Θ=arc cos[(A.B)/(|A||B|)]

I understand that there are 2 components. I trying to outline the idea first.
Suppose, if I can show that theta does not change, would I then have proven?

 

[BvU]

Right! the dot product divided by the magnitudes should give you something constant. From now on, it is math all the way to the answer.

 

[BvU]

Yes. Showing (A.B)/(|A||B|) is constant is sufficient.
Now, the exercise is written out in polar coordinates. You can do a lot of work transforming to cartesian, or you can stay with polar (r, Θ).

 

[negation]

Yes. Showing (A.B)/(|A||B|) is constant is sufficient.
Now, the exercise is written out in polar coordinates. You can do a lot of work transforming to cartesian, or you can stay with polar (r, Θ).

I'm a little disturbed by the amount of hassle having to make the conversion from polar to cartesian and vice versa. It's tedious and messy. Do you recommend learning lagrangian in first year physics or should I really focus on the existing math and physics in school first?

 

[BvU]

Focus. But $x=r \cos\theta$ and $y = r \sin\theta$ is first year. So is the dot product in polar coordinates $(r_1, \theta_1)\cdot (r_2, \theta_2) = |r_1|\, |r_2|\, \cos(\theta_1 - \theta_2)$

There is no way to avoid math in phys, I would say. Philosophers might dream otherwise.