The velocity of a heavy meteorite entering the earth's...

The velocity of a heavy meteorite entering the earth's atmosphere is inversely proportional to $\sqrt{s}$ when it is $s$ kilometers from the earth's center. Show that the meteorite's acceleration is inversely proportional to $s^{2}$.

Key Concepts

Chain Rule in Differentiation

The chain rule is used in calculus to differentiate composite functions. In the context of the problem, it allows us to differentiate the velocity function with respect to s and then relate it back to time by recognizing that ds/dt is the velocity, thereby linking the derivatives dv/ds and dv/dt.

Relationship between Velocity and Acceleration

Acceleration is defined as the rate of change of velocity with respect to time. By applying the chain rule to the velocity function given, the derivative gives an acceleration that is inversely proportional to s^2, which demonstrates how the gravitational influence on the meteorite changes with distance from the center of the earth.

Inverse Proportionality

This concept means that one quantity varies as the reciprocal of another. In the problem, the velocity being inversely proportional to the square root of s implies that an increase in s results in a decrease in velocity according to the relation v(s) = k/?s, where k is a constant.

Transcript

00:02 In this question, where we are talking about chain rule, we are given that a heavy meteorite is entering earth's atmosphere with a velocity that is inversely proportional to the square root of s, where s is the distance of the meteorite from earth center.

00:25 In this question, we want to show that the acceleration of the meteorite is inversely proportional to s squared.

00:35 Okay, so where do we start? well, since we're interested in the acceleration, that's where i want to start.

00:46 What is acceleration? it is the time rate of change of velocity, whose expression we can deduce from the given information.

00:58 Now, we don't know what velocity equals, but we do know that it is inversely proportional to s squared.

01:10 And so, do we know what it equals? well, we know that proportional proportion when a quantity is proportional to another, it is always a constant multiple of the other.

01:26 So if my paycheck or something is inversely proportional to my costs or somebody else's paycheck or whatever, i know mine will always be a constant multiple of the other, no matter what to the other one might happen to be.

02:02 So in this case, in this example that i'm writing here, this top number is proportional to the bottom number by the proportionality constant 2.

02:18 And so that means the velocity is just a scalar multiple of 1 over the square root of x.

02:26 Now that we have that equality, we can differentiate it with respect to t.

02:36 Since k is a constant, it is just a constant multiple on the derivative.

02:44 And this derivative is of 1 over the square root of us.

02:58 Now, we're differentiating with respect to time, and this is not a function of time, or not directly.

03:07 It is, however, a function of the distance, which changes with time because the meteorite is following.

03:16 Thus, we need to use the chain rule...

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