A drone ascends vertically while simultaneously moving north

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

A drone ascends vertically while simultaneously moving north. A ground receiver situated 200 m east of the drone’s launch point tracks the drone by rotating an antenna. At time

t

seconds, the drone’s horizontal displacement north of its launch point is

p(t)

meters and its altitude is

h(t)

meters. The line of sight from the receiver to the drone forms an angle

\theta(t)

(in radians) measured upward from the horizontal plane. The geometry implies

\theta(t)=\arctan\!\left(\frac{h(t)}{\sqrt{p(t)^{2}+200^{2}}}\right).

Suppose that at

t=30

s the drone is 120 m north of the launch point and 80 m above ground, rising at 3 m s

^{-1}

and flying north at 6 m s

^{-1}

. (a) Derive a simplified formula for

\dfrac{d\theta}{dt}

in terms of

h,\,p,\,\dfrac{dh}{dt},

and

\dfrac{dp}{dt}

. (b) Evaluate

\dfrac{d\theta}{dt}

t=30

s. Provide the exact value, then a decimal approximation to four decimal places (units: rad s

^{-1}

). (c) The receiver’s motor can rotate the antenna at a maximum angular speed of 0.050 rad s

^{-1}

. Will the antenna be able to keep tracking the drone at

t=30

s? Justify your answer numerically.

Related theory

Browse the problem library