Concept explainers
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
where
(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
(c) Use calculus to obtain the closed form solution where
(d) Use Euler's method to obtain a numerical solution from
(a)
The differential equation for velocity v if the gravitational force is not constant and is given by the function
Answer to Problem 12P
Solution:
The differential equation for velocity v is
Explanation of Solution
Given Information:
The function
Where,
Assume the upward velocity is positive. Therefore, the force balance is given as,
Here,
And,
And,
Thus, the force balance is,
Divide the both sides of the above equation by m,
Hence, the differential equation for the velocity is,
(b)
The differential equation of for velocity as a function of altitude if the differential equation for velocity as a function of time is,
Answer to Problem 12P
Solution:
The differential equation of
Explanation of Solution
Given Information:
The differential equation
The drag force is negligible.
And, the chain rule is given as,
Consider the chain rule,
Here,
Now, consider the equation,
Since, drag force is negligible. Therefore,
Thus, from the chain rule,
Hence, differential equation of velocity as a function of altitude x is
(c)
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
Answer to Problem 12P
Solution:
The solution for the velocity is,
Explanation of Solution
Given Information:
The differential equation
The initial condition, at
Formula used:
Integration formula,
Calculation:
Consider the differential equation,
Separate the variable as below,
Integrate both the sides of the above equation,
Now, for
Substitute the value of C in the equation
Hence, the solution for the velocity is
(d)
To calculate: The velocity from
Answer to Problem 12P
Solution:
The velocity from
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
Where,
The initial condition
Formula used:
Euler’s method for
Where, h is the step size.
Calculation:
Consider the differential equation,
Substitute
The iteration formula for Euler’s method with step size
From part (c), the analytical solution for the velocity is,
Substitute
Use excel to find all the iteration with step size
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
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
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
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.
Want to see more full solutions like this?
Chapter 1 Solutions
EBK NUMERICAL METHODS FOR ENGINEERS
Additional Engineering Textbook Solutions
Basic Technical Mathematics
Advanced Engineering Mathematics
Fundamentals of Differential Equations (9th Edition)
McDougal Littell Jurgensen Geometry: Student Edition Geometry
The Heart of Mathematics: An Invitation to Effective Thinking
- For the given velocity triangle, if you know that at the inlet, alpha is -20, beta is 59, the relative velocity is 78 m/s, r= 20 cm, rotation speed is 180 rpm. At the exit, alpha is 30 and the absolute velocity is 35 m/s. Find the following: 1. the relative velocity at the exit in (m/s) 2. the work in (KJ/Kg) Vw1 U, V1 V1 V2 Uz Center of wheelarrow_forwardPide Use Buckingham's PI Theorem to determine non-dimensional parameters in the phenomenon shown on the right (surface tension of a soap bubble). The variables involved are: R AP - pressure difference between the inside and outside R- radius of the bubble Pide Soap film surface tension (Gravity is not relevant since the soap bubble is neutrally buoyant in air)arrow_forwardAn object of mass m is being affected by the force F = cxx, where c is a positive constant. The object leaves the origin with a speed i, at timet = 0. Find x(x). Answer to Question 1 [Insert solution image after this line]: %3D %3Darrow_forward
- m, zm 2. Two particles of equal masses move on a frictionless harizontal surface. Their center of kr? mass is fixed and their potential energies are krž and Also, they interact with each other with potential 2 akr2 Ris the distance between the masses, and k and a are positive constants. a. Find the Lagrangian in terms of the center of mass position CM, R and the relative position r. b. Solve the Lagrange equations for the relative coordinates X,Y and x,y. c. Explain the physical outcome of the results obtained in b.arrow_forward2. A moving car, represented as a point, has a time-varying position given by z(t) = a +be² -ct and a time-varying velocity given by (1)=2b1²-6ct² where a = 2.17 m. b = 4.80 m/s², and c= 0.100 m/s. Find: (a) the car's average velocity from 2.00 to 10.0 s, (b) the car's average acceleration during the same interval. (c) the car's velocity at t = 5.0 s. (d) and the car's acceleration at the same time.arrow_forwardThe small particle of mass (800 gm) spinning with angular velocity (w₁ = 2.3 rad/s, at r₁ = 400 mm), if we use force (F) to change the Angular velocity to (w₂, and r₂ = 250 mm) the value of F is: W1 Select one: A. None B. F = 5.63 N C. F = 9.47 N D. F = 3.52 N Farrow_forward
- Tp = Fq +°P/Q• (1) Here ip/Q is the "position of point P relative to point Q." Similarly the velocities of the two points are related by õp = bq + Up/Q- (2) The quantity õp/Q is the velocity of point P relative to point Q. I want you to use these ideas to solve the following problems. 1. The figure below shows a view from above of a large boat in the middle of the ocean. So that the crew on the ship can get exercise on long journeys, there is a circular walking/running track on the back deck. CA B- -D Suppose that the radius of the track is R = 6 m, and a person is running on the track at a constant speed of v = 3m/s as measured with a stopwatch by a crew-mate on board the ship. Suppose the runner is running counter-clockwise around the track when viewed from above. Write the velocity vector of the runner in terms of basis (ê1, ê2) as perceived by a crew-mate on the ship. (a) What is the velocity vector when the runner is at point A? (b) What is the velocity vector when the runner is…arrow_forwardQ1: A100 kg man in space throws a 20kg space rock. As a result, the man accelerates to the left at 1m/s2.what is the acceleration of the rock?arrow_forwardA particle rotates with constant speed in a circle. Let be the net torque on the particle and F the net force on the particle. Then: a. tau > 0 and F > 0 O b. tau > 0 and F = 0 . tau = 0 and F = 0 Od. tau = 0 and F > 0arrow_forward
- Kinetic energy of a fluid flow can be computed by 1 pv · vdV, where p(x, y, z) and v(x, y, z) are the pointwise fluid density and velocity, respectively. Fluid with uniform density flows in the domain bounded by x2 + z² = 6 and 0 < y<. = The velocity of parabolic flow in the given domain is v(x, y, z) 6 (6 – x² – 2²)j. Find the kinetic energy of the fluid flow.arrow_forwardDamian’s car weighs 2000kg. The spring has a natural unstretched length of 2m and a spring constant of k = 80000N/m. Hooke's law can be written as T = kd. Where T is the tension force in newtons, k is the spring constant and d is the length in metres. Let x(t) be the position of the front of Damian’s car and let y(t) be the position of the back of Eva’s 4WD. We will assume that the position of Eva’s car is a known function of time. Q1 a) Create a sketch of the positions of the vehicles similar to the one given and add the positions x and y. b) What is the extension of the spring in terms of x(t) and y(t)? Be careful to take into account that the unstretched length of the spring is 2m.arrow_forwardFor a venturi meter given below, the volumetric flow rate is defined in terms of the geometrical parameters, the density of working fluid (p), and density of the manometer liquid (pm) as 4. Q = f(D, D2, A2, g, h, Pmv Pr) %3D Write down the balance equations and show your work to end up with an expression for the volumetric flow rate in terms of the variables defined above.arrow_forward
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY