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Assignment
1. Root-finding: Bisection
An underground tank for storing diesel has the shape of a cylinder laying on its side. The (horizontal) length of the cylinder is L, and its (vertical) radius is R. The volume of diesel inside the container when the level is h from the bottom is given by
V (h) = L �R2 cos−1 �R − h� − (R − h) √2Rh − h2� R
(A) Use the formula to find the volume of diesel in the tank when h = 0, R, & 2R, and check using the regular formula for the (total) volume of a cylinder (V = πR2L) that the above formula is correct for these special values of h.
(B) The owner of the tank wants to order more diesel when the volume of diesel in the tank drops below 10% of the total volume, at which point the height from the bottom will be below h10. Find the condition for the value of h10 for which the owner should place an order when the level drops below this value. Write this condition in the form
f(h) = 0 if the tank dimensions are L = 10m and R = 3m.
(C) Use the bisection method to find h10 to 2 decimal places. Write your answer in a tabular form similar to page 40 of the type-set notes. Take the initial values for h in the bisection method to be a0 = 0 and b0 = R.
2. Root-finding: Newton-Raphson
There are two locations on the interval 0 < x < 4 where the function
intersects with the function
At these points f(x) = g(x) − h(x) = 0.
g(x) = x3 − 4.4x2 + 4.05x + 11.5 h(x) = 10 sin2(2x)
(A) Find the derivative of f(x) with respect to x.
(B) Use the Newton-Raphson method to find the two solutions to f(x) = 0 for 0 < x < 4. Perform the method for the following 2 initial guesses x0 = {2,3} and iterate for 10 steps unless f(xn) drops below 10−10, in which case you have converged.
(C) You should have found that for each of the 2 starting points converges to the same root. This is one of the pitfalls of the Newton-Raphson method. Plot the function f(x) for 0 < x < 4 (try desmos.com), and by looking where it crosses y = 0 choose a appropriate starting value (with no more than 2 significant figures) so that the algorithm converges to the second root.
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3. ODE-solver Application The orbit of a asteroid or satellite around the Sun can be com- puted by solving ordinary differential equations for the position and velocity. Such an orbit will lie in a plane, which we assume to be the xy-plane. The object will feel a force due to gravity of
r2
where rˆ is a unit vector pointing out radially from the Sun. This force results in an accel-
�
eration �a = F/m by Newton’s second Law. We can thus describe the motion of the object
using the following set of ODEs:
dx
= vx = vy
= ax = − = ay = −
GM◦ m F=− rˆ
�
dt dy
dt dvx
dt dvy
GM◦ x r3 GM◦ y r3
Sun is M◦ = 1.98847 × 1030 kg.
To apply Euler’s method (pg 44 of the typeset notes) to solve for the orbit, we use the
following algorithm based on an appropriate small step in time h:
dt
where the gravitational constant is G = 6.67408 × 10−11 m3 kg−1 s−2, and the mass of the
(i) Start with given values for xn, yn, vx,n, vy,n. Compute rn = (ii) Compute the derivative functions:
k1x = hvx,n , k1y = hvy,n , k1vx = −hGM◦xn r3
(iii) Step along the orbit to the next point using
xn+1 = xn + k1x yn+1 = yn + k1y vx,n+1 = vx,n + k1vx vy,n+1 = vy,n + k1vy
�22 xn + yn.
,
k1vy = −hGM◦yn r3
(iv) Return to step (i) and repeat with the new values obtained in step (iii).
I have included a sample MATLAB (or Octave) code that implements Euler’s method for this problem. I have also created a sample spreadsheet that implements Euler’s method in case you do not have access to any other form of programming language. In what follows we will make use of “astronomical units”, where 1 AU = 1.495978707 × 1011 m is the average distance between the Earth and the Sun.
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Questions:
(A) Plot the orbit (i.e. x versus y) for Euler’s method for the conditions:
� GM◦ r0
where r0 = 1 AU. Run the code for 250 steps (i.e. a total time of 500 days). By considering the physical parameters used in this problem, comment on the accuracy of the resulting orbit. (HINT: just take my code or spreadsheet, increase the number of steps from 100 to 250 and capture the plot. Also, compare these parameters to those of Earth’s orbit around the Sun).
(B) Adapt either the MATLAB code or the spreadsheet to use the second order Runge- Kutta (RK2) method (also called the “Improved Euler” or “Cauchy Euler” method) from Chapter 5, Section 2 (pg 46) of the typeset notes.
(C) Run your RK2 method for the same conditions as the Euler method I provided (i.e. those listed in part A) for 250 steps. Plot the resulting orbit on the same axes as the one you obtained for Euler’s method (or side by side on axes with the same scale). Comment on the accuracy of the RK2 method compared to Euler’s method.
(D) Determine the approximate amount (∼ 2 significant figures is fine) that the trajectory deviates from 1 AU after one orbit (i.e. the “error”) of the RK2 method with h = 1 day, then repeat this for h = 2 days and h = 4 days. The quantity to compute here is the relative error ((|r − r0|)/r0). Comment on the results.
• BONUS (an additional 5% of the assignment total): Extend your code to employ the RK4 method. Repeat the analysis of the error from part (D) for your RK4 method. Comment on the results.
x0 =r0 , y0 =0, vx,0 =0, vy,0 =
, h=2×60×60×24
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