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(a) The Empire State Building is 1250 feet high. With what minimal velocity must a ball be thrown vertically upward to reach its top. (b) If an object is dropped from a height of 1250 feet, with what speed does it hit the ground?(c) What is the relationship between the answers to the questions and why isit so?

(a) $282.84 \mathrm{ft} / \mathrm{sec}$(b) $282.84 \mathrm{ft} / \mathrm{sec}$(c) gravity acts the same way in both cases.

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 2

Applications of Antidifferentiation

Integrals

Missouri State University

Campbell University

Oregon State University

University of Nottingham

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

05:30

A ball is dropped from the…

02:40

04:10

01:05

Gravity If a ball is dropp…

04:11

A ball is thrown verticall…

01:08

A ball is thrown straight …

03:50

06:23

02:28

a ball is dropped from the top of the empire state building and after t seconds. This is its height right here in feet. Okay. The first question is find the velocity and what's the sign of the velocity and why? Well, we can answer the last two. Before we do any work. The ball is falling to the ground. So the velocity is negative because the velocity has a size and a direction. So the negative means down or to the left or to the right, whichever way is negative in your problem. So sure up is positive. Down this negative. Okay, Final velocity. The velocity is the derivative of the height, So -32. T. Okay. How do I know for sure? That's negative. Well t is time and time is always positive or zero? Okay. So so at the beginning the velocity is zero but after one second it's negative. All right. Find the acceleration and discuss it. That isn't exactly what they said, but that's what we'll do. The acceleration is the velocity the derivative of the velocity which is the second Derivative of the height or the position. So it's -32. Mhm. Back to this one. It was in feet the height was in feet we took its derivative with respect to time which is in seconds. So it's in feet per second. Okay, the last one was in feet per second and we took as derivative with respect to seconds. So it's feet per second per second. AR -32 ft/s Squared. Mhm. All right. So it's a constant uh sign is negative. It's units are feet per second squared. Um I don't know what else they want to know about it. Okay. How many seconds till it hits the ground? Well, you don't set it equal the velocity equal to zero. The velocity of zero outward hits the ground, but when it hits the ground we don't know what its velocity is. So we have to get the velocity. Um or we have to get the time it takes to hit the ground from the original equation. Remember why is the height? Okay. And when t is zero? It's 12 50 That's the top of the Empire State Building. Okay. When T0, we're up here at 12:50. So when it hits the ground, what's the height? Well the height zero there this was a this is B. C. So you need to know when does 1250 minus 16 T squared equal to zero? So minus 16 T squared equals 12 50 minus 12 50. So t squared is 1250 over 16. So T is the square root of 1250 over 16 seconds. Let me see if that's something worth simplifying 1250 divided by 16. Um I'll just go ahead and calculate it but I'll Uh leave calculate with that that's only 8.83 seconds Round it off. Oh a four sorry Okay. What's its velocity then? Well we'll plug it into the velocity Equation which remember was -32 T. So lost equals -32 times The square to 1250 over 16. Yeah. Mhm Times 32 2 82 .8 ft/s. Hm. All right. Oh then it also wants me to change that to MPH so to 8 to 8 feet per second. Times 15/22 MPH Per one ft per second. All right. 2828. I still have all the decimals online, so times 15 Divided by 22, wow I got 192.8 mph. I would not have guessed that.

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