Apex Calculus, Chapter 4, Section 2, Excercise 7
An F-22 aircraft is flying at 500mph with an elevation of 10,0000 on
a straight–line path that will take it directly over an anti–aircraft
gun. How fast must the gun be able to turn to accurately track the
aircraŌ when the plane is:
(a) 1 mile away?
(b) 1/5 mile away?
(c) Directly overhead?
\[\frac{dx}{dt}=500 mph=733.33
ft/s\]
\[cotan (\theta )=
\frac{x}{10,000}\]
\[\frac{d}{dt}(cotan (\theta ))=
(\frac{x}{10,000})\frac{d}{dt}\]
\[-csc^{2} (\theta ) \frac{d\theta}{dt}=
(\frac{1}{10,000})\frac{dx}{dt}\]
\[\frac{d\theta}{dt}=
(\frac{-sin^{2}\theta}{10,000})\frac{dx}{dt}\]
theta <- atan(10000/5280)
#Answer (a)
(-(sin(theta))^2/10000)*-733.33## [1] 0.05734588#Answer (b)
theta <- atan(10000/(5280/5))
(-(sin(theta))^2/10000)*-733.33## [1] 0.07252426#Answer (c) - I cannot calculate directly above the camera since the horizontal lenght is zero. I approximate the speed at 1ft away from the camera
theta <- atan(10000/(1))
(-(sin(theta))^2/10000)*-733.33## [1] 0.073333