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Chapter 3: Q3-3.4-14E (page 116)

An object of mass mis released from rest and falls under the influence of gravity. If the magnitude of the force due to air resistance is bvn, where band nare positive constants, find the limiting velocity of the object (assuming this limit exists). [Hint:Argue that the existence of a (finite) limiting velocity implies that dvdt→0as t→+∞

Short Answer

Expert verified

The limiting velocity isv=(mgb)1n .

Step by step solution

01

Find the limiting velocity of the object

According to the newton’s second law of equation mdvdt=mg-bvn

02

Apply given conditions

Letdvdt=0and t→+∞

0=mg-bvnv=(mgb)1n

Therefore, the limiting velocity is role="math" localid="1663965910386" v=(mgb)1n.

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Most popular questions from this chapter

The solution to the initial value problemdydx=(x+y+2)2,y(0)=-2crosses the x-axis at a point in the interval [0,14].By experimenting with the improved Euler’s method subroutine, determine this point to two decimal places.

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.


Determine the recursive formulas for the Taylor method of order 4 for the initial value problem y'=x2+y,y(0)=0.

An object of mass 5 kg is given an initial downward velocity of 50 m/sec and then allowed to fall under the influence of gravity. Assume that the force in newtons due to air resistance is -10v, where v is the velocity of the object in m/sec. Determine the equation of motion of the object. If the object is initially 500 m above the ground, determine when the object will strike the ground.

Use the improved Euler’s method with tolerance to approximate the solution to y'=1-y+y3,y(0)=0, at x= 1. For a tolerance ofε=0.003 , use a stopping procedure based on the absolute error.
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