It is common to see birds of prey rising upward on thermals. The paths they take may be spiral-like. You can model the spiral motion as uniform circular motion combined with a constant upward velocity. Assume a bird completes a circle of radius 8.00m every 5.00s and rises vertically at a rate of 3.00 m/s .

Question

Find the speed of the bird relative to the ground.

Find the magnitude of the bird's acceleration

Find the direction of the bird's acceleration.

Find the angle between the bird's velocity vector and the horizontal.

Can someone please help with the forumla's?

Answer 1

The vertical motion and the horizontal motion are decoupled. The vertical velocity is v = 3 m/s and the vertical acceleration is 0 m/s^2
In the horizontal plane the speed is u = 2 pi r/T = 2 pi (8/5) = 10 m/s
and the acceleration is centripetal = v^2/r = 12.6 m/s^2 toward the center of the spiral
So the speed relative to ground is:
s = sqrt(u^2+v^2)
The acceleraation is all inward in the horizontal plane and is 12.6 m/s^2
tan theta = v/u = 3/10

Answer 2

given R, T and Vy (vertical velocity component).
A)
find horizontal velocity component
Vx=2πR/T
this is the horizontal component, use the pythagorean theorem with the vertical velocity given to find net speed: Vx^2 (we just found)+Vy^2(given upward velocity of bird)= V^2 (dont forget to sqrt V)
B)A=V^2/R
C) It's 0.
D) using the Vx and Vy we already found, we need to use trig to find the angle between the adjacent and the hypotenuse, so we use tan(theta)=opposite/adjacent, in this case opposite being vertical Vy and the adjacent being horizontal Vx.
So arctan(Vy/Vx)= theta.

gig 'em