Spiraling Up. It is common to see birds of preyrising upward on thermals. The pa

Question

Spiraling Up. It is common to see birds of preyrising upward on thermals. The paths they take may be spiral-like.You can model the spiral motion as uniform circular motion combinedwith a constant upward velocity. Assume a bird completes a circleof radius 6.00m every 5.00s and rises vertically at a rate of 3.00 m/s.

(a)Find the speed of the bird relative to the ground.

(b) Find the magnitude of the bird's acceleration.

(c) Find the direction of the bird's acceleration.

(d) Find the angle between the bird's velocity vector and thehorizontal.

Explanation / Answer

(a) the bird moves with vertical speed of 3.00 m/s and horizontal (circular) speed of

distance / time = 2 pi r / T = 2 * pi* 6.00 / 5.00 = 7.54 m/s

so his speed relative to the ground is the magnitude of his velocity vector. To get this, we use the pythagorean theorem (imagine drawing a triangle with vertical leg 3.00 and horizontal leg 7.54... the hypotenuse is the birds total speed) or:

speed rel to ground = sq root (3.00*3.00 + 7.54*7.54) = 8.11 m/s

(b) the bird has zero acceleration vertically, and his horiz acc is due to circular motion, so...

mag of acc = v^2 / r = 7.54*7.54 / 6.00 = 9.47 m/s^2

(c) the direction of the birds acceleration is always horizontal, toward the center of the spiral (because there is no vertical component of acceleration)

(d) angle the birds velocity vector makes with the horizontal... again, think of the triangle from part b:

     angle = arctan (3.00 / 7.54) = 21.7 degrees

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