Question: An object with mass is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If s(t)

An object with mass is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If s(t) is the distance dropped after t seconds, then the speed is v = s(t) and the acceleration is a = v '(t). If g is the acceleration due to gravity, then the downward force on the object is mg - cv, where c is a positive constant, and Newton's Second Law gives
m dv / dt = mg - c v
(a) Solve this as a linear equation to show that
v = mg / c (1 - e-ct / m)
(b) What is the limiting velocity?
(c) Find the distance the object has fallen after t seconds.

Step by Step Solution

★★★★★

3.35 Rating (164 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a dv dt c m v g and It e c m dt e c mt and multiplying the differential equation by ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

786-C-D-E (683).docx

120 KBs Word File

Students Have Also Explored These Related Calculus Questions!

A force with magnitude 20 N acts directly upward from the xy-plane on an object with mass 4 kg. The object starts at the origin with initial velocity v(0) = i - j. Find its position function and its...

(a) A lamina has constant density p and takes the shape of a disk with center the origin and radius R. Use Newtons Law of Gravitation (see Section 13.4) to show that the magnitude of the force of...

The position of an object with mass m at time t is r(t) = at 2 i + bt 3 j, 0

b. How far has each object travelled at time t = 3 s? c. What is the slope of each line? A 20 g mass and a 50 g mass are dropped from rest from the same height above the oon a. If we ignore air...

Some questions may have more than one correct answer. . Let C (t) be the concentration of a drug in the bloodstream. As the body eliminates the drug, C(t) decreases at a rate that is proportional to...

PY CH6 HW2:- 5) A planet of mass 4.00 x 1025 kg is in a circular orbit of radius 4.00 x 1.011 m around a star. The star exerts a force on the planet of constant magnitude 1.34 x 1023 N. The speed of...

Homework # 15 - Velocity and Distance with Acceleration Handout Problems: 1. A youngster throws a ball vertically upward. As the ball rises to its highest point and then falls, what changes, if any,...

Could you please help with the following question? part(b) and (c) please give the detailed solving process. How do you integrate? and how to take limit and prove it's E.S 1. For this entire problem,...

1. In which one of the following cases is the displacement of the object linearly proportional to the elapsed time? O a ball at rest is given a constant acceleration O a ball rolls with constant...

Refer to Exercises 95 and 96. a. Compute a jumpers terminal velocity, which is defined as b. Find the terminal velocity for the jumper in Exercise 96 (m = 75 kg and k = 0.2). c. How long does it take...

Anne Sorenson, an auditor for Stuart and Cram, is currently assigned to the audit of Amgen, Inc., a biotechnology company based in Thousand Oaks, California. The audit has gone well, but today...

On December 31, 2007, Drew Company issued $170,000, five-year bonds for $155,000. The stated rate of interest was 6 percent and interest is paid annually on December 31. Required: Provide the...

CSE 1 2 1 - - Introduction to " C " Programming Assignment # 8 Due: _ see canvas Name your program: p 8 . c This week we will write some code which will output text to a file ( instead of to the...

Old Faithful? Does the duration of an eruption have an effect on the length of time that elapses before the next eruption? a) The histogram below shows the duration (in minutes) of those 222...

Evaluate the limit. e2x e-2x lim In(x + 1)

Produce graphs of f that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease and intervals of concavity, and use calculus to find these intervals exactly. 2...

Find dy/dx by implicit differentiation. x + y = x 4 + y 4

Local Co . has sales of $ 1 0 . 2 million and cost of sales of $ 6 . 1 million. Its selling, general and administrative expenses are $ 5 1 0 , 0 0 0 and its research and development is $ 1 . 2...

The Macaulay duration of a 10 year $100 par value bond with annual coupons of $X is 6.96 years at an annual effective interest rate of 10%. Calculate X. Give your answer as a decimal rounded to two...

The Cutting Department of Cullumber Company has the following production and cost data for July. Production Costs 1. Completed and transferred out 10,080 units. Beginning work in process $0 2. 2,520...