Analyzing Bee's Spiral Path: Velocity & Acceleration

[chaotixmonjuish]

1.)

A bee goes from its hive in a spiral path given in plane polar coordinates by
r = b*e^kt , θ = ct,
where b, k, c are positive constants. Show that the angle between the velocity vector and the
acceleration vector remains constant as the bee moves outward. (Hint: Find v · a/va.)

so here is my v and a

2.)
v = (r')e_r+(r*θ')e_θ

a = (r''+rθ')e_r+(rθ''+2r'θ')e_θ

r' = bk*e^kt
r'' = bk²e^kt
θ' = c

3.) my attempt at a solution

bke^kt(bk²e^kt-be^ktc²)+(be^ktc)(2bke^ktc)

is that the right dot product

this is where I'm stuck

 

[chaotixmonjuish]

so I got down to this:

e^2kt(a bunch of constants)+e^2kt(a bunch of constants)/e^4kt


all the e's canceled out and left just constants

 

[markpace123]

My answer is for the dot product is:

e^2kt(b²k³ + b²kc + 2b²kc²)

 

[markpace123]

you continue by finding the modulus of v and a then by using the hint you cross out the

b²e^2kt and end up with cos^-1(a bunch of constants) and the answer