Falling meteorite The velocity of a heavy met...

Question

Falling meteorite The velocity of a heavy meteorite entering Earth’s atmosphere is inversely proportional to √s when it is s km from Earth’s center. Show that the meteorite’s acceleration is inversely proportional to s².

Accepted Answer

Welcome back, everyone. A satellite falls directly toward a planet from a large initial distance. Its velocity is inversely proportional to the cube root of its distanced from the planet's center. How is the acceleration of the satellite related to D? A says the acceleration of the satellite is directly proportional to the 5/3 power of its distanced from the planet's center. B says directly proportional to the 4/3 power. C inversely proportional to the 5/3 power, and D inversely proportional to the 4/3 power. So essentially what we're going to do is define our function. It says that our satellite falls directly toward a planet from a large initial distance and its velocity V is inversely proportional to the cube root of its distance, so we can include a proportionality constant K. And because we have an inverse proportionality, we're going to observe a fractional relationship, V equals K divided by cube root of distance D, right? Now, what we have to understand is that V is a function of distance, and distance is a function of time, right? So, essentially, when we Calculate the acceleration. Which is the first derivative of velocity, we will have to apply the chain rule. So let's differentiate the velocity function. The derivative of velocity is going to be equal to the acceleration. This is exactly what we're looking for, and basically what we have to do is differentiate V. With respect to D, so we're calculating DV divided by DD and according to the chain rule we're multiplying by the derivative of distance with respect to time. We also know that the first derivative of distance or position with respect to time is velocity, so we get our final relationship as DV divided by DD multiplied by V itself, right? So let's go ahead and evaluate the derivative of velocity. With respect to distance, and we can do that using the power rule because essentially we have k multiplied by the race to the power of -1/3 using the reciprocal rule and the rules of exponents. We can rewrite the cube root as 1/3, right? The negative sign comes from the fact that we have our term in the denominator. So here we are applying the reciprocal rule. Now we're going to get K multiplied by 1/3 based on the power rule. Multiplied by the raises the power of -1/3 minus 1, which is negative for 3. And now if we use this expression in our acceleration. We're going to get acceleration equals -13 K. D is the power of -4/3 multiplied by V, right? And our V is K. Multiplied by the. To the power of negative 1/3. Using the properties of exponents, we can essentially multiply those terms. First of all, we are going to have -13. Multiplied by k squared. And now applying the properties of exponents, we're going to get D to the power of -4/3 plus 1/3, which is -5/3. And now we can rewrite it as a negative key squared. Divided by 3. D It's the power of 5/3. So what we can tell is that our relationship is an inverse proportionality because we have distance in the denominator and the exponent is 5/3. So essentially considering the answer choices, we're looking for 5/3 and we are looking for inversely proportional, which means that the correct answer to this problem corresponds to the answer choice C. Thank you for watching.

Falling meteorite The velocity of a heavy meteorite entering Earth’s atmosphere is inversely proportional to √s when it is s km from Earth’s center. Show that the meteorite’s acceleration is inversely proportional to s².

Key Concepts

Inverse Proportionality

Differentiation

Chain Rule

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