Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Suppose an object is dropped from a plane flying at a constant altitude.(a) First, suppose that the only force on the object is gravity, determine its impact velocity. (b) Now suppose that in addition to gravity, there is a frictional force proportional to the velocity opposing the object's motion, that is, its acceleration is $a=g-k v$ where $k$ is a constant. Determine the velocity of this object and show that the velocity will approach $g / k$ as $t$ gets very large. This is called the terminal velocity.

(a) $\sqrt{-2 g s(0)}$(b) $v=\frac{g\left(1-e^{-k t}\right)}{k}$

03:26

Zulfiqar A.

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 2

Applications of Antidifferentiation

Integrals

Campbell University

Oregon State University

University of Michigan - Ann Arbor

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.

03:19

An object falling under th…

11:42

A parachutist having a mas…

05:12

According to one model tha…

02:39

A falling object is said t…

02:21

01:14

05:41

If a unit mass is dropped …

12:25

An object with mass $ m $ …

this problem on the topic of Newton's laws of motion, we have an object that is falling under the influence of gravity and also acted upon by frictional force of air resistance. If we know the magnitude of this force is approximately proportional to the speed of the object so that the frictional force committed as B times V, where we are given the constant B and the mass of the object, we want to find the terminal speed that the object reaches while falling. And then we want to know if this answer depends on the initial speed of the object. So here we have a force diagram showing the frictional force acting upward in the object, which is minus B V and its weight acting vertically downward, which is M Times G, where Emma's its mess. Now the object will fall so that, according to Newton's second Law, M. A is equal to the weight mg minus the frictional Resistance BV, meaning that if we rearrange the equation, we get the acceleration. A to B mg minus b V over M. M is the mass of this object. If we take the downward direction to be positive, we know equilibrium. We reached when a is equal to zero. So when a is equal to zero, we get the velocity to be terminal velocity and this is simply MG divided by B and this is the mass of the object. 50 kg times the acceleration due to gravity 9.8 meters per square second divided by B, which is 15 kgs per second, which gives us the terminal velocity of this object to be 33 meters per second. So this is the speed which the object cannot exceed. Now we want to know if the speed depends on the initial velocity. Now, if the initial velocity we not is less than 33 m the second, then we have a positive acceleration a greater than zero. But we still have a final velocity the of 33 meters per second. That's the largest velocity that the object can attain. On the other hand, if the initial velocity we not is greater than 33 m a second, then we have a negative exhilaration or a deceleration. And 33 m per second is the smallest velocity attained by the object. But it's still the final velocity or the terminal velocity attained by the object. So even if the initial velocity is 33 m per second, acceleration is zero and the object continues to fall with the constant speed of 33 m per second. So the answer is no. Our answer in part a does not depend on the initial velocity.

View More Answers From This Book

Find Another Textbook

Numerade Educator

05:30

Evaluate the given integral.$$\int x y \sqrt{1-x^{2}-y^{2}} d x$$

01:57

(a) determine a definite integral that will determine the area of the region…

01:54

$$\begin{aligned}&\text { If } f(x, y, z)=4 x^{2}+2 y^{3}+5 z^{5}+3 …

02:46

Use the properties about odd and even functions to evaluate the given integr…

02:17

$$\begin{aligned}&\text { If } f(x, y, z)=\ln \left(x^{2}+y^{2}+z^{2…

01:56

Evaluate the given integral and check your answer.$$\int\left(5 x^{2}+2 …

02:10

Sketch the area represented by the given definite integral.Integral in E…

01:24

Sketch the area represented by the given definite integral.$$\int_{1}^{2…

02:15

Determine $G(x)$.$$G(x)=\int_{1}^{x} t^{4} \sqrt[3]{1-t^{2}} d t$$

01:44

Evaluate the given integral and check your answer.$$\int\left(t^{2}+1\ri…