Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 3: Q9E (page 116)

An object of mass 100kg is released from rest from a boat into the water and allowed to sink. While gravity is pulling the object down, a buoyancy force of 1/40 times the weight of the object is pushing the object up (weight = mg). If we assume that water resistance exerts a force on the object that is proportional to the velocity of the object, with proportionality constant 10N-sec/m , find the equation of motion of the object. After how many seconds will the velocity of the object be 70 m/sec ?

Short Answer

Expert verified

The result is x (t)=95.65t-956.5e-t10-956.5 and time taken by the object is 13.2 sec .

Step by step solution

01

Important hint.

Use Newton’s method to solve for t tn+1=tn-f(tn)f'(tn).

02

Find the equation of motion

The total force acting on the object is F-mg-bv-140mg.

Applying newton’s second law:

100dvdt=100 (9.81)-10 v-1  (100)  (9.81)40

v'=9.56-0.1vv'+0.1v=9.56v'+0.1v=9.56               onsolvingbyvariableseparating

When v (0)=0,  then  C=-95.65.

v=95.65-95.65e-t10x(t)=95.65t-956.5e-t10+c …… (1)

When x(0)=0,  then  C=-956.5.

x (t)=95.65 t-956.5 e-t10-956.5

03

Find the value of t

When the object travelling at the velocity 70m/sec then by equation (1).

70=95.65 t-956.5 e-t1070=95.65(1-e-t10)t=13.2 sec

Therefore, the result is x (t)=95.65 t-956.5 e-t10-956.5 and t=13.2 sec.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A 400-lb object is released from rest 500 ft above the ground and allowed to fall under the influence of gravity. Assuming that the force in pounds due to air resistance is -10V, where v is the velocity of the object in ft/sec determine the equation of motion of the object. When will the object hit the ground?

Use the improved Euler’s method subroutine with step size h= 0.1 to approximate the solution to the initial value problemy'=x=y2,y(1)=0, at the points x= 1.1, 1.2, 1.3, 1.4, and 1.5. (Thus, input N= 5.) Compare these approximations with those obtained using Euler’s method (see Exercises 1.4,Problem 5, page 28).

A garage with no heating or cooling has a time constant of 2 hr. If the outside temperature varies as a sine wave with a minimum of 50°Fat2:00a.m.and a maximum of80°Fat2:00p.m., determine the times at which the building reaches its lowest temperature and its highest temperature, assuming the exponential term has died off.

Local versus Global Error. In deriving formula (4) for Euler’s method, a rectangle was used to approximate the area under a curve (see Figure 3.14). With

\({\bf{g(t) = f(t,f(t))}}\), this approximation can be written as \(\int\limits_{{{\bf{x}}_{\bf{n}}}}^{{{\bf{x}}_{{\bf{n + 1}}}}} {{\bf{g(t)dt}} \approx {\bf{hg(}}{{\bf{x}}_{\bf{n}}}{\bf{)}}} \)where \({\bf{h = }}{{\bf{x}}_{{\bf{n + 1}}}}{\bf{ - }}{{\bf{x}}_{\bf{n}}}\) .

  1. Show that if g has a continuous derivative that is bounded in absolute value by B, then the rectangle approximation has error\(\left( {\bf{O}} \right){{\bf{h}}^{\bf{2}}}\); that is, for some constant M, \(\left| {\int\limits_{{{\bf{x}}_{\bf{n}}}}^{{{\bf{x}}_{{\bf{n + 1}}}}} {{\bf{g(t)dt - hg(}}{{\bf{x}}_{\bf{n}}}{\bf{)}}} } \right| \le {\bf{M}}{{\bf{h}}^{\bf{2}}}\).This is called the local truncation error of the scheme. [Hint: Write \(\int\limits_{{{\bf{x}}_{\bf{n}}}}^{{{\bf{x}}_{{\bf{n + 1}}}}} {{\bf{g(t)dt - hg(}}{{\bf{x}}_{\bf{n}}}{\bf{)}}} {\bf{ = }}\int\limits_{{{\bf{x}}_{\bf{n}}}}^{{{\bf{x}}_{{\bf{n + 1}}}}} {\left[ {{\bf{g(t)dt - g(}}{{\bf{x}}_{\bf{n}}}{\bf{)}}} \right]{\bf{dt}}} \). Next, using the mean value theorem, show that\(\left| {{\bf{g(t)dt - g(}}{{\bf{x}}_{\bf{n}}}{\bf{)}}} \right| \le {\bf{B}}\left| {{\bf{t - }}{{\bf{x}}_{\bf{n}}}} \right|\) . Then integrate to obtain the error bound\(\left( {\frac{{\bf{B}}}{{\bf{2}}}} \right){{\bf{h}}^{\bf{2}}}\).]
  2. In applying Euler’s method, local truncation errors occur in each step of the process and are propagated throughout the further computations. Show that the sum of the local truncation errors in part (a) that arise after n steps is (O)h. This is the global error, which is the same as the[ss1] [m2] convergence rate of Euler’s method.


During the summer the temperature inside a van reaches 55°C, while that outside is a constant35°C. When the driver gets into the van, she turns on the air conditioner with the thermostat set at16°C. If the time constant for the van is1k=2 hrand that for the van with its air conditioning system is1k1=13 hr, when will the temperature inside the van reach 27°C?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free