ODE -What happened to the moth?

[cyt91]

ODE --What happened to the moth?

One theory about the behaviour of moths states that they navigate at night by keeping fixed angle between their velocity vector and the direction of the Moon [or some bright star]. A certain moth flies near to a candle and mistakes it for the Moon. What will happen to the moth?
Hints: in polar coordinates (r,θ ), the formula for the angle ω between the radius vector and the velocity vector is given by

Use the formula to solve for r as a function of .

I do not know how to set up the necessary equations. I tried sketching a diagram,but I cannot seem to find a relation between ω and θ necessary to solve the formula given. Is my diagram correct? Thank you for your help.

 

[lippyka]

Yes, your diagram is correct. In order to solve the formula given, you need to use the trigonometric identities for cosθ and sinθ. Then, you can isolate r and find a relation between ω and θ. For example, you can use the identity cos2θ + sin2θ = 1 to solve for cosθ in terms of sinθ, and then substitute it into the formula. This will give you an equation relating ω and θ.