Package: Loose Leaf For Numerical Methods For Engineers With 1 Semester Connect Access Card
Package: Loose Leaf For Numerical Methods For Engineers With 1 Semester Connect Access Card
7th Edition
ISBN: 9781259289163
Author: Steven C. Chapra Dr., Raymond P. Canale
Publisher: McGraw-Hill Education
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Textbook Question
Chapter 1, Problem 12P

In our example of the free-falling parachutist, weassumed that the acceleration due to gravity was a constant value. Although this is a decent approximation when we are examining falling objects near the surface of the earth, the gravitational force decreases as we move above sea level. A more general representation based on Newton's inverse square law of gravitational attraction can be written as

g ( x ) = g ( 0 ) R 2 ( R + x ) 2

where g ( x ) = gravitational acceleration at altitude x (in m) measured upward from the earth's surface ( m/s 2 ) ,   g ( 0 ) = gravitational acceleration at the earth's surface ( 9.81  m/s 2 ) , and R = the earth's radius ( 6.37 × 10 6  m ) .

(a) In a fashion similar to the derivation of Eq. (1.9) use a force balance to derive a differential equation for velocity as a function of time that utilizes this more complete representation of gravitation. However, for this derivation, assume that upward velocity is positive.

(b) For the case where drag is negligible, use the chain rule to express the differential equation as a function of altitude rather than time. Recall that the chain rule is

d v d t = d v d x d x d t

(c) Use calculus to obtain the closed form solution where v = v 0  at  x = 0 .

(d) Use Euler's method to obtain a numerical solution from x = 0 to 100,000 m using a step of 10,000 m where the initial velocity is 1500 m/s upward. Compare your result with the analytical solution.

(a)

Expert Solution
Check Mark
To determine

The differential equation for velocity v if the gravitational force is not constant and is given by the function g(x)=g(0)R2(R+x)2.

Answer to Problem 12P

Solution:

The differential equation for velocity v is dvdt=g(0)R2(R+x)2+Cdv|v|m.

Explanation of Solution

Given Information:

The function g(x)=g(0)R2(R+x)2.

Where, g(x) is the gravitational acceleration at an altitude x, g(0) is the acceleration due to gravity at the surface of earth which is 9.81 m/sec2, R is the earth’s radius which is 6.37×106 m.

Assume the upward velocity is positive. Therefore, the force balance is given as,

F=Fgravity+Fdrag

Here,

F=ma=mdvdt

And,

Fgravity=mg(x)=mg(0)R2(R+x)2

And,

Fdrag=Cdv|v|

Thus, the force balance is,

mdvdt=mg(0)R2(R+x)2+Cdv|v|

Divide the both sides of the above equation by m,

dvdt=mg(0)mR2(R+x)2+Cdv|v|m=g(0)R2(R+x)2+Cdv|v|m

Hence, the differential equation for the velocity is, dvdt=g(0)R2(R+x)2+Cdv|v|m.

(b)

Expert Solution
Check Mark
To determine

The differential equation of for velocity as a function of altitude if the differential equation for velocity as a function of time is, dvdt=g(0)R2(R+x)2+Cdv|v|m.

Answer to Problem 12P

Solution:

The differential equation of dvdt for the special case of a sphere is dvdt=g(1ρρs)3ρCdv|v|4ρsd.

Explanation of Solution

Given Information:

The differential equation dvdt=g(0)R2(R+x)2+Cdv|v|m.

The drag force is negligible.

And, the chain rule is given as,

dvdt=dvdxdxdt

Consider the chain rule,

dvdt=dvdxdxdt

Here, dxdt=v is the velocity. Therefore,

dvdt=dvdxv

Now, consider the equation,

dvdt=g(0)R2(R+x)2+Cdv|v|m

Since, drag force is negligible. Therefore,

dvdt=g(0)R2(R+x)2+0=g(0)R2(R+x)2

Thus, from the chain rule,

dvdt=dvdxvg(0)R2(R+x)2=dvdxvdvdx=g(0)vR2(R+x)2

Hence, differential equation of velocity as a function of altitude x is dvdx=g(0)vR2(R+x)2.

(c)

Expert Solution
Check Mark
To determine

To calculate: The solution for the velocity by the use of calculus if the differential equation of velocity as a function of altitude x is dvdx=g(0)vR2(R+x)2.

Answer to Problem 12P

Solution:

The solution for the velocity is, v=±2g(0)(RxR+x)+v02.

Explanation of Solution

Given Information:

The differential equation dvdx=g(0)vR2(R+x)2.

The initial condition, at x=0, v=v0.

Formula used:

Integration formula,

xndx=xn+1n+1+C

Calculation:

Consider the differential equation,

dvdx=g(0)vR2(R+x)2

Separate the variable as below,

vdv=g(0)R2(R+x)2dx

Integrate both the sides of the above equation,

vdv=g(0)R2(R+x)2dxv22=g(0)R2dx(R+x)2v22=g(0)R2(1(R+x))+Cv22=g(0)(R2(R+x))+C

Now, for x=0, v=v0. Therefore, the above equation is,

v022=g(0)(R2(R+0))+Cv022=g(0)R+CC=v022g(0)R

Substitute the value of C in the equation v22=g(0)(R2(R+x))+C as below,

v22=g(0)(R2(R+x))+v022g(0)Rv22=g(0)(R2(R+x)R)+v022v2=2g(0)(R2R2RxR+x)+v02v=±2g(0)(RxR+x)+v02

Hence, the solution for the velocity is v=±2g(0)(RxR+x)+v02. Here, the positive sign represent upward direction and negative sign indicates downward direction.

(d)

Expert Solution
Check Mark
To determine

To calculate: The velocity from x=0 to 100,000 m by the Euler’s method with the step size Δx=10,000 m and the initial condition v(0)=1500 m/sec if the differential equation of velocity as a function of altitude x is dvdx=g(0)vR2(R+x)2. Also, compare the result with the analytical solution.

Answer to Problem 12P

Solution:

The velocity from x=0 to 100,000 m by the Euler’s method is,

x v- Euler v- analytical
0 1500 1500
10000 1434.518 1433.216
20000 1366.261 1363.388
30000 1294.818 1290.023
40000 1219.669 1212.476
50000 1140.138 1129.885
60000 1055.324 1041.05
70000 963.9789 944.2077
80000 864.2883 836.5811
90000 753.4434 713.3028
100000 626.6846 564.2026

The velocity by the Euler’s method is approximately same as the analytical solution.

Explanation of Solution

Given Information:

The differential equation dvdx=g(0)vR2(R+x)2.

Where, g(x) is the gravitational acceleration at an altitude x, g(0) is the acceleration due to gravity at the surface of earth which is 9.81 m/sec2, R is the earth’s radius which is 6.37×106 m.

The initial condition v(0)=1500 m/sec.

Formula used:

Euler’s method for dydx=f(x,y) is,

y(x+h)=y(x)+hf(x,y)

Where, h is the step size.

Calculation:

Consider the differential equation,

dvdx=g(0)vR2(R+x)2

Substitute g(0)=9.81 m/sec2 and R=6.37×106 m in the above equation and solve as below,

dvdx=9.81v(6.37×106)2(6.37×106+x)2=398.56×1012(6.37×106+x)2v

The iteration formula for Euler’s method with step size Δx=10,000 m for the above equation is,

v(x+10,000)=v(x)+10,000(398.56×1012(6.37×106+x)2v)

From part (c), the analytical solution for the velocity is,

v=±2g(0)(RxR+x)+v02

Substitute g(0)=9.81 m/sec2, and R=6.37×106 m in the above equation and solve as below,

v=±2×9.81×(6.37×106x6.37×106+x)+(1500)2

Use excel to find all the iteration with step size Δx=10,000 m from x=0 to 100000 m as below,

Step 1: Name the column A as x and go to column A2 and put 0 then go to column A3and write the formula as,

=A2+10000

Then, Press enter and drag the column up to the x=100000 m.

Step 2: Now name the column B as v-Euler and go to column B2 and write 1500 and then go to the column B3 and write the formula as,

=B2+10000*(-398.56*10^12/((6.37*10^6+A2)^2*B2))

Step 3: Press enter and drag the column up to the x=100000 m.

Step 4. Now name the column C as v-analytical and go to column C2 and write 1500 and then go to the column C3 and write the formula as,

=(-2*9.81*((6.37*10^6*A3)/(6.37*10^6+A3))+1500^2)^(1/2)

Step 5. Press enter and drag the column up to the x=100000 m.

Thus, all the iterations are as shown below,

x v- Euler v- analytical
0 1500 1500
10000 1434.518 1433.216
20000 1366.261 1363.388
30000 1294.818 1290.023
40000 1219.669 1212.476
50000 1140.138 1129.885
60000 1055.324 1041.05
70000 963.9789 944.2077
80000 864.2883 836.5811
90000 753.4434 713.3028
100000 626.6846 564.2026

To draw the graph of the above results, follow the steps as given below,

Step 6:Select the column A and column B. Then, go to the Insert and select the scatter (X, Y) from the chart.

Step 7: Select the column A and column C. Then, go to the Insert and select the scatter (X, Y) from the chart.

Step 8: Select one of the graphs and paste it on another graph to Merge the graphs.

The graph obtained is,

From the graph, it is observed that both the graphs of velocity by analytical method and by Euler’s method is approximately same.

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Chapter 1 Solutions

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