Question

Related rates: an aircraft is flying at 460 ft/s with an elevation of 100 ft from ground. how fast...?

An aricraft is flying at 460 ft/s with an elevation of 100 ft from the ground, on a straight-line path that will take it directly over an anti-aircraft gun. How fast, in radians per second, will the gun have to turn to accurately track the aircraft when the plane is:

1000 ft away?
100 ft away?

Answer 1

At a distance of a 1000 ft, the gun turns at `~~0.0455` rad/s, while at a distance of 100 ft, it turns at `2.3` rad/s

Explanation:

Let `x` be the distance of the aircraft from the point on its path directly vertically above the AA gun. We denote the height of the aircraft by `h`, and the angle that the gun makes with the horizontal as it track the aircraft by `theta`.

It is easy to see that

`x = h cot theta`

and so

`dx/dt = -h csc^2 theta\ (d theta)/(dt) implies`

`(d theta)/dt = -1/h sin^2 theta\ dx/dt = -1/h h^2/(x^2+h^2) dx/dt`
`qquad quad = - h/(x^2+h^2) dx/dt`

Now, in the problem it is given that `h=100\ "ft"` and `dx/dt = -460\ "ft/s"` (note the minus sign - this indicates that `x` is decreasing as the aircraft approaches the gun). So

• @ `x=1000\ "ft"`
  `(d theta) /dt = -(100\ "ft")/((100\ "ft")^2+(1000\ "ft")^2) times (-460\ "ft/s")`
  `qquad ~~0.0455\ "s"^-1`
• @ `x=100\ "ft"`
  `(d theta) /dt = -(100\ "ft")/((100\ "ft")^2+(100\ "ft")^2) times (-460\ "ft/s")`
  `qquad ~~2.3\ "s"^-1`