Question

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 6.00 m every 5.00 s and rises vertically at a rate of 3.00 m/s. Find the angle between the bird's velocity vector and the horizontal

Answer 1

Radius of circle of spiral path = 6 m

Time period = 5 s

So the total length of the path = $2 \pi R$

$distance = 2 \pi R$

$distance = 2 \pi *6$

$distance = 12\pi$

time taken by bird to cover the distance = 5 s

so the speed of the bird = distance / time

$v = \frac{distance}{time}$

$v = \frac{12\pi}{5}$

$v = 7.54 m/s$

so the tangential speed in horizontal direction = 7.54 m/s

vertical velocity by which it is rising upwards = 3 m/s

so the angle with the horizontal for net speed is given as

$\theta = tan^{-1}\frac{v_y}{v_x}$

$\theta = tan^{-1}\frac{3}{7.54}$

$\theta = 21.7 degree$

so velocity vector will make 21.7 degree with the horizontal