[40 points] Imagine you dug a tunnel straight through the center of the earth to the...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
[40 points] Imagine you dug a tunnel straight through the center of the earth to the other side, and jumped in. You will experience acceleration due to gravity (g), but this will decrease as you approach the center of the planet (ignore the molten core) and then start increasing again as you start to move to the surface on the other side. You will also experience a drag force proportional to your velocity in the direction opposite your motion. The governing equation of motion is expressed by the following ODE for velocity: m dv dt msign(r)g(r) sign(v)cav, where vis velocity (m/s); t = time (s); g(r) is the changing gravitational constant (m/s) as a function of height above the Earth's center (r, in m); sign(x) is a function that returns -1, 0, and 1 for negative, zero, and positive values of x, respectively (you can use the Python function numpy.sign() for this); ca is the drag coefficient (kg/m); and m is mass (kg). Acceleration due to gravity is given by where G = 6.674 x 10-11 mkgs average density of the Earth. 4 3 g(r) = Gp is the gravitational constant and p = 5515 kg/m is the In both cases, calculate the position and velocity of the jumper for t = 0 to 3 x 105s (a bit over 80 hours) given the following parameters: m = 65.8 kg, and cd = 0.25 kg/m. Assume that the initial height is the radius of the Earth, 6.371 x 100 m, and zero initial velocity. Hint: Start by writing the equation in terms of position. Could we maybe turn that into..a system? a) Try using the midpoint method (i.e., modified Euler method) with a step size of 0.5 s. (Perhaps just solve to an end time of 100 s.) Can you obtain a reasonable solution? What does this tell you? b) Now solve this using scipy.integrate.solve_ivp(). Plot the resulting position and velocity as a function of time. Interpret the behavior-does this seem reasonable? Explain. Also, what is the range of time-step sizes that the integrator chose? [40 points] Imagine you dug a tunnel straight through the center of the earth to the other side, and jumped in. You will experience acceleration due to gravity (g), but this will decrease as you approach the center of the planet (ignore the molten core) and then start increasing again as you start to move to the surface on the other side. You will also experience a drag force proportional to your velocity in the direction opposite your motion. The governing equation of motion is expressed by the following ODE for velocity: m dv dt msign(r)g(r) sign(v)cav, where vis velocity (m/s); t = time (s); g(r) is the changing gravitational constant (m/s) as a function of height above the Earth's center (r, in m); sign(x) is a function that returns -1, 0, and 1 for negative, zero, and positive values of x, respectively (you can use the Python function numpy.sign() for this); ca is the drag coefficient (kg/m); and m is mass (kg). Acceleration due to gravity is given by where G = 6.674 x 10-11 mkgs average density of the Earth. 4 3 g(r) = Gp is the gravitational constant and p = 5515 kg/m is the In both cases, calculate the position and velocity of the jumper for t = 0 to 3 x 105s (a bit over 80 hours) given the following parameters: m = 65.8 kg, and cd = 0.25 kg/m. Assume that the initial height is the radius of the Earth, 6.371 x 100 m, and zero initial velocity. Hint: Start by writing the equation in terms of position. Could we maybe turn that into..a system? a) Try using the midpoint method (i.e., modified Euler method) with a step size of 0.5 s. (Perhaps just solve to an end time of 100 s.) Can you obtain a reasonable solution? What does this tell you? b) Now solve this using scipy.integrate.solve_ivp(). Plot the resulting position and velocity as a function of time. Interpret the behavior-does this seem reasonable? Explain. Also, what is the range of time-step sizes that the integrator chose?
Expert Answer:
Related Book For
Physics
ISBN: 978-0077339685
2nd edition
Authors: Alan Giambattista, Betty Richardson, Robert Richardson
Posted Date:
Students also viewed these mechanical engineering questions
-
Imagine that a hole is drilled through the center of the Earth to the other side. An object of mass m at a distance r from the center of the Earth is pulled toward the center of the Earth only by the...
-
In Problem 19, what is a differential equation for the displacement x(t) if the motion takes place in a medium that imparts a damping force on the spring/ mass system that is proportional to the...
-
Graph each inequality or compound inequality. 5x - y > 6
-
Samples of six different brands of diet/imitation margarine were analyzed to determine the level of physiologically active polyunsaturated fatty acids (PAPFUA, in percentages), resulting in the...
-
Discuss the ethical issues that have developed regarding the use of IT in cross-border transactions. What new conflicts have developed since the printing of this book? What solutions can you suggest?
-
Texas Inpatient Consultants, LLLP, is a partnership that employs physicians to deliver medical care to hospitalized patients of other physicians. Texas Inpatient recruited Julius Tabe, M.D., to work...
-
Fenton Company applied FIFO to its inventory and got the following results for its ending inventory . Cameras 100 units at a cost per unit of ....... $68 DVD players 150 units at a cost per unit of...
-
Bank extends a revolving line of credit to Epstein Corp., secured by Epstein's inventory and receivables. Epstein has now filed bankruptcy. At all times during the 90 days preceding the filing the...
-
? Dean and Ellen Price are married and have a manufacturing business. They bought a piece of business equipment (7-year personal property) on 4/1/2017 for $50,000. Use half-year convention to...
-
Develop a detailed checklist below prioritising the installation of the new networking components, with an estimated time for each task. Add new rows as necessary./ The networking components will...
-
What is the primary difference between the corporate costs of capital for investor-owned and not-for-profit firms?
-
Should the component cost estimates reflect historical or marginal costs?
-
Mr. Girdhari Lal does not keep full double entry records. His balance as on January 01, 2006 is as. He withdrew Rs. 500 per month out of which to spent Rs. 1,500 for business purpose. Prepare the...
-
What are some roles that investment bankers play in mergers?
-
What is the general formula for the corporate cost of capital?
-
1) Calculate Norton equations in polar form. 2) Draw the equivalent circuit of Norton. 3) To make charge work with maximum power, what should be the values to connect circuit.(Use polar and complex...
-
QUESTION 9 HC-O-C-R R-C-O-CH HC-O-P-O-CH-CH-NH3* O || O a. Phosphatidic acid, Serine O b. Lysophosphatidic acid, Serine, Free FA O c. Lysophosphatidylserine, Free FA O d. 2 Free FAs, Serine, Glycerol...
-
Near sea level, how high a hill must you ascend for the reading of a barometer you are carrying to drop by 1.0 cm Hg? Assume the temperature remains at 20 C as you climb. The reading of a barometer...
-
It is a damp, chilly day in a New England seacoast town suffering from a power failure. To warm up the cold, clammy sheets, Jen decides to fill hot water bottles to tuck between the sheets at the...
-
The weight of an object at the surface of a planet is proportional to the planet's mass and inversely proportional to the square of the radius of the planet. Jupiter's radius is 11 times Earth's and...
-
In the past five years, there have been significant innovations in technology such as smartphones and tablets. Technology companies rely on intellectual property (IP) rights, such as patents,...
-
Respondents Varsity Brands, Inc., Varsity Spirit Corporation, and Varsity Spirit Fashions & Supplies, Inc., design, make, and sell cheerleading uniforms. Respondents have obtained or acquired more...
-
Mr. Tam is the front man for the Asian American dance-rock band The Slants. Mr. Tam named his band The Slants to reclaim and take ownership of Asian stereotypes. The band draws inspiration for its...
Study smarter with the SolutionInn App