When a rocket is launched, it is tracked by a tracking dish...

When a rocket is launched, it is tracked by a tracking dish on the ground located a distance $D$ from the point of launch. The dish points toward the rocket and adjusts its angle of elevation $\theta$ to the horizontal (ground level) as the rocket rises. Suppose a rocket rises vertically at a constant speed of $2.0 \mathrm{~m} / \mathrm{s}$ with the tracking dish located $150 \mathrm{~m}$ from the launch point. Find the rate of change of the angle $\theta$ of elevation of the tracking dish with respect to time $t$ (tracking rate) for each of the following: (a) Just after launch. (b) When the rocket is $100 \mathrm{~m}$ above the ground. (c) When the rocket is $1.0 \mathrm{~km}$ above the ground. (d) Use the results in (a)-(c) to describe the behavior of the tracking rate as the rocket climbs higher and higher. What limit does the tracking rate approach as the rocket gets extremely high?

When a rocket is launched, it is tracked by a tracking dish on the ground located a distance $D$ from the point of launch. The dish points toward the rocket and adjusts its angle of elevation $\theta$ to the horizontal (ground level) as the rocket rises. Suppose a rocket rises vertically at a constant speed of $2.0 \mathrm{~m} / \mathrm{s}$ with the tracking dish located $150 \mathrm{~m}$ from the launch point. Find the rate of change of the angle $\theta$ of elevation of the tracking dish with respect to time $t$ (tracking rate) for each of the following:
(a) Just after launch.
(b) When the rocket is $100 \mathrm{~m}$ above the ground.
(c) When the rocket is $1.0 \mathrm{~km}$ above the ground.
(d) Use the results in (a)-(c) to describe the behavior of the tracking rate as the rocket climbs higher and higher. What limit does the tracking rate approach as the rocket gets extremely high?

Key Concepts

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to time given the rate of change of another related quantity. In these problems, multiple quantities are tied together through equations or geometrical relationships, and the goal is to differentiate these relationships with respect to time to find unknown rates.

Trigonometric Relationships in Right Triangles

Trigonometric relationships, such as using the tangent function in a right triangle, are used to relate angles to side lengths. In problems involving angles of elevation, the tangent of the angle is defined as the ratio of the opposite side (e.g., height) to the adjacent side (e.g., horizontal distance), providing a useful bridge between geometry and calculus.

Implicit Differentiation

Implicit differentiation is a method used to differentiate equations that define one variable implicitly in terms of another, rather than explicitly. In the context of related rates, it allows us to differentiate both sides of an equation with respect to time, applying the chain rule as needed, to connect the rates of change of different variables.

Differentiation of Inverse Trigonometric Functions

Differentiating inverse trigonometric functions, such as the arctan function, is essential when the variable of interest (in this case, an angle) is expressed as an inverse trigonometric function of other quantities. Understanding these derivatives helps in obtaining the rate at which the angle changes.

Asymptotic Behavior and Limits

Analyzing limits involves determining the behavior of a function as the input grows without bound or approaches a particular value. In related rates problems, understanding the asymptotic behavior of the tracking rate as the rocket ascends to very high altitudes can illustrate how the rate of change of the angle approaches a limiting value.

Transcript

00:01 So a rocket is launched from this point and it's going up at a rate of two meters per second.

00:13 So there is a dish or a tracking dish that is located 150 meters from the launch line.

00:21 So let's say our tracking dish is right over here.

00:24 So the distance here is 150 meters.

00:30 Okay, so we're looking for the rate of change of the angle of elevation from the distance.

00:36 A tracking dish to the rocket with respect to time.

00:41 So angle would be if we connect a tracking dish to our rocket.

00:48 So theta is this angle right over here.

00:53 And let's call this distance here y.

01:00 Then we can see that we need to write an equation that relates theta and y.

01:09 So let's write that what we have.

01:11 So we know that.

01:12 At d .y d .t is equal to two.

01:19 And what we're looking for is we want to find d theta d t at different, i guess at different distances for y.

01:35 So the first thing we're going to do before doing the different parts is let's write on the equation that relates beta and y.

01:41 So we're going to use tangent theta because we have our opposite, that is y.

01:50 And we have our adjacent because that is just a constant.

01:54 It's always 150 meters.

01:58 Okay, so this tells us that since we want d -theta d -t, we want to take the derivative with respect to t on both sides of this equation.

02:06 And if we do that, we get second squared theta times d -t is equal to 1 over 150 times d -y -d -t.

02:21 Okay, so meaning if we're we're looking for d theta d t that is just equal to uh so this is one over 150 sequin square theta which we can actually write as cosine theta since secan theta is one over cosine so this is really co squared theta over 150 times d y d t okay so now we're ready to do the different part a is we are looking for d -feta d -t just after launch.

03:04 So meaning we're looking for d -feta -d -t evaluated at y is equal to 0.

03:15 So if y is equal to 0, what is our cosine square theta? well, cosine is adjacent over hypotenics.

03:28 So if y is equal to zero, then theta is basically zero here.

03:35 Right.

03:35 So that means cost of zero is one.

03:40 Okay, so this is one over 150 times dydt, which is two.

03:46 So this is one over 75.

03:50 And our units here would be radians per second.

03:58 Now part to b, we are looking for d theta, dt, and this time we're evaluating this when the rocket is 100 meters above the ground.

04:10 So when y is equal to 100.

04:16 Okay, so then this is equal to.

04:18 So again, we're looking for cosine square theta.