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A rocket is launched vertically upward with an initial velocity of $6400 \mathrm{ft} / \mathrm{sec}$(a) When will the velocity of the rocket be zero? (b) What is the maximum altitude the rocket will attain?A ball is thrown vertically upward from the ledge of a building 256 feet high with an initial velocity of $16 \mathrm{ft} / \mathrm{sec}$. What is the ball's impact velocity with the ground?A helicopter is stationary at an altitude of 512 feet. A package falls vertically

(a) $576 \mathrm{ft}$(b) $12 \sec$(c) -208 ft/sec

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 7

Marginal Functions and Rates of Change

Derivatives

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

05:14

A rocket is shot verticall…

01:04

If a toy rocket is launche…

05:35

A toy rocket is launched (…

12:45

A model rocket blasts off …

04:39

Use the height equation in…

03:14

A rocket blasts off vertic…

07:52

A catapult launches a test…

09:34

08:27

A rocket rises vertically,…

11:54

A test rocket is fired ver…

08:09

04:47

A test rocket is fired ve…

04:27

A toy rocket fired straigh…

07:13

A model rocket is launched…

Mhm. So the maximum height of the rocket we thought first. Accelerating height maxim had the toy is the first accelerating height. Let's say it is extra and then the engine feels, Let's say it has traveled to Hide H. two. So the maximum height it's max is the height when the toy is accelerating and the hardest to when it is accelerating under the gravitation of uh so the high maximum height H max is H one plus H. Two. They can write x minus. That's when it's given. There is 1000 m. And as to is the velocity of the toy at the end of the Phase one. That is even spare upon to into the gravitational expression. Okay, this is the maximum attachments. Now let's calculate the velocity we once were. So we have even square is equal story than usual. Speed. You square Plus two into the acceleration a into the height edge. That's one. Okay, so can we even square is you're zero Plus two. Into the airport acceleration is 13. We just for a second for second square into the height. H one is 1000 ft or even square is uh 60,000. It was 60,000 meters square four seconds square. What said This is a question two and this is a question one. So from the question one and two. The maximum height H max is 1000 m for each one. Yes, 60,000. We just square for 2nd square upon Uh, to into the value of G. That is 9.8 mi. Just four seconds square. Or the maximum height. It maxes sleep was to 4061 in Egypt. This is a solution for part how parties. The total time tears spiralled off to edge upon the acceleration in the face. One to H one upon acceleration is faced one plus two and to the height H two in the first two. And the exploration in the face to that is cheap. The heart is too in the face tools 4061 m -1000 m, or the height H two is 3061 m. So the total time taken to reach the maximum height is uh two and 2 1000 metre Upon 13. We just for a second, square in the square root Plus Square root of 20 3061 m upon 9.8 metre four seconds square. Or the total time to reach the maximum. Itis 33.2 sects, no partisan. So when the when the toy reached the maximum height, it falls for a time period of the 53 of 0.5 seconds. Okay, so the distance traveled uh By the toy will be on the downward direction and that is equals two As three is the question. Half of G into the T. three sq or we can say its trees half of 9.8 into 0.5 metre 3.5 square meter or Goddess threes mm 1.23 m. And the velocity at the end when the Parachute uh explodes is equals two G Into the time. T. three and that is equals two 9.8 into 0.5 m for a second. On the street with the parachute explores is 4.9 me just for a second. Yeah, so the total time He is equals two. The time to reach the maximum height that is 33.2 seconds Plus the free fall time. That is 0.5 seconds plus the Hi descending. With the uniform speed. That is 4,061 -1.23 m Upon the speed, that is 4.9 m/s of the total time. Our flight is 8/62 seconds.

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