Homework 1 - Due Sep 19th

Due Sep 19, 2019 by 5p.m.
Points 20

Problems in red will be graded.

Problems in Lynch, Section 2.6 -- 1 (a), (d), (e), 5 (a), (b), and 10 (only (a) will be graded)
Find two different solutions to the initial value problem
$\dot{x}=2\sqrt{|x|}\cos t ,\ x(0)=0.$
Can you find infinitely many different solutions to the same initial value problem?
Consider the initial value problem

$\dot{x}=\frac{y}{z}$

$\dot{y}=-\frac{x}{z}$

$\dot{z}=1$

with $(x(1), y(1), z(1)) = (1, 0,1).$

Convert this system to cylindrical coordinates $(r,\theta,z)$ where $r=\sqrt{x^2+y^2}$ and $\theta = \tan^{-1}(y/x)$ . Find the initial conditions in the new coordinate system. Solve the new system and show that its solution exists in the maximal interval $(0,\infty)$ .
Apply Picard iteration for the initial value problem $\dot{x}= x^2,\ x(0) = 1$ and obtain a series solution. Find the interval where the series solution converge. (Hint: What is the radius of convergence of the power series?) What elementary function has the Taylor expansion (around 0) that agrees with the obtained series solution?
This year we celebrated the 50th anniversary of the Apollo 11 landing on the moon. Let's explore some physics related to this human achievement. Apollo 11 should have been launched at the escape velocity from earth, i.e., the initial velocity of a projectile moving vertically upwards starting from the earth's surface so that it escapes earth's gravitational pull.

a) Calculate this escape velocity using the energy methods you learned in the course: First, set up the equation of motion for the rocket using Newton's second law, and then, figure out the Energy preserved by the motion. Finally, in order to calculate the escape velocity use the fact that having escaped the gravity, the rocket is in motion (positive velocity).

b) Find the time $t_M$ Apollo 11 would have taken to reach the moon, assuming it was launched with the escape velocity. Use that the distance from the surface of the Earth to the Moon is $d = 405 696\ km$ . The actual time Apollo 11 took to reach moon's orbit was 75 hours and 56 minutes. Does this agree with the answer you found? If not, can you explain why?

--- Recall that the acceleration of gravity at the surface of Earth is $g = \frac{GM}{R^2} = 9.81\ m/s^2$ , and that the radius of Earth is $R = 6378\ km$ . Here $M$ denotes the mass of the Earth, and $G$ is Newton’s gravitational constant.

--- The force on such a mass $m$ when is at a distance $y$ from the surface of the Earth is, according to Newton’s gravitational law, is $\frac{GMm}{(R + y)^2}$ . With this you should be able to write down the equation of motion using Newton's second law.

--- Note that the escape velocity is independent of the mass of the object.