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A helicopter is stationary at an altitude of 512 feet. A package falls vertically from it. Determine the time it takes the package to hit the ground and its impact velocity with the ground.

$$5.66 \mathrm{sec},-181.02 \mathrm{ft} / \mathrm{sec}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 7

Marginal Functions and Rates of Change

Derivatives

Oregon State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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.

02:57

(II) A helicopter is ascen…

01:53

01:43

04:44

A helicopter 8.50 m above …

02:39

A hot-air balloon is ascen…

02:47

A balloon is ascending at …

02:38

Free-Fall Acceleration…

03:16

02:35

A helicopter is flying in …

03:19

An airplane with a speed o…

in this question. We have an object being dropped off of a helicopter. So I've gone had enlisted my given and I've also assigned my positive direction. Now, remember, in physics you get to choose which direction to be positive. Uh, I solved this one, assuming that up is the positive direction. Okay, so the helicopter was traveling upward at 5.20 meters per second. So that's why it is a positive number and the object is dropped from a height of 125 meters. So that makes it my initial position. And my final position is gonna be zero meters and then again because, um, it's falling. It's in free fall. So the acceleration is gonna be acceleration due to gravity, which is now a negative 9.8 meters per second. Because my positive direction is the upward direction. Okay, Now, on a problem like this, it's very important to keep your directions straight because essentially what's happening is the object first goes up and then it's gonna have an equal amount of time to return to the starting point, and then it's gonna fall the rest of the way down to the ground. Okay, so you really want to make sure that you keep your directions straight on this one. Now, um, the easiest way to do this problem are the kind of most direct way with the fewest steps is to actually use the quadratic formula for acceleration. So we have the final position is equal to 1/2 a T squared, plus initial velocity times, time plus initial position. And then we can just plug everything in. So 1/2 negative, 9.8 times time. Let's times time squared. Don't forget to square it and then 5.2 times time plus 125. And that's all equal to zero. Now, To solve the quadratic pick your favorite way you can factor. You can use an online quadratic calculator. You can use your calculator. Um, you know, maybe you've got the fancy kind that will solve the quadratic formula for you. What? However you want to do it, we need to solve our quadratic formula. And when we do, we find that the time for that object to reach the ground is going to be five 0.61 seconds. Okay,

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