3.6: Problems

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Use Euler’s Method to determine the given value of \(y\) for the following problems. When possible compare the numerical approximations with the exact solutions.
a. \(\dfrac{d y}{d x}=2 y, y(0)=2 .\) Find \(y(1)\) with \(h=0.1\)
b. \(\dfrac{d y}{d x}=x-y, y(0)=1 .\) Find \(y(2)\) with \(h=0.2\)
c. \(\dfrac{d y}{d x}=x \sqrt{1-y^{2}}, y(1)=0 .\) Find \(y(2)\) with \(h=0.2\).
d. \(\dfrac{d y}{d t}=1+\dfrac{y}{t}, y(1)=2\) with \(h=0.25\).
e. \(\dfrac{d y}{d t}=-3 y+t e^{2 t}, y(0)=0\) with \(h=0.25\).
Use the Midpoint Method to solve the initial value problems in Problem \(1 .\)
Numerically solve the nonlinear pendulum problem using the EulerCromer code for a pendulum with length \(L=0.5 \mathrm{~m}\) using initial angles of \(\theta_{0}=10^{\circ}\), and \(\theta_{0}=70^{\circ} .\) In each case run the routines long enough and with an appropriate \(h\) such that you can determine the period in each case. Compare your results with the linear pendulum period.
For the Baumgartner sky dive we had obtained the results for his position as a function of time. There are other questions which could be asked.
1. Find the velocity as a function of time for the model developed in the text.
2. Find the velocity as a function of altitude for the model developed in the text.
3. What maximum velocity is obtained in the model? At what time and position?
4. Does the model indicate that terminal velocity was reached?
5. What speed is predicted for the point at which the parachute opened?
6. How do these numbers compare with reported data?
Consider the flight of a golf ball with mass \(46 \mathrm{~g}\) and a diameter of \(42.7\) \(\mathrm{mm}\). Assume it is projected at \(30^{\circ}\) with a speed of \(36 \mathrm{~m} / \mathrm{s}\) and no spin.
1. Ignoring air resistance, analytically find the path of the ball and determine the range, maximum height, and time of flight for it to land at the height that the ball had started.
2. Now consider a drag force \(f_{D}=\dfrac{1}{2} C_{D} \rho \pi r^{2} v^{2}\), with \(C_{D}=0.42\) and \(\rho=1.21 \mathrm{~kg} / \mathrm{m}^{3}\). Determine the range, maximum height, and time of flight for the ball to land at the height that it had started.
3. Plot the Reynolds number as a function of time. [Take the kinematic viscosity of air, \(v=1.47 \times 10^{-5}\).
4. Based on the plot in part \(c\), create a model to incorporate the change in Reynolds number and repeat part b. Compare the results from parts \(a, b\) and \(d\).
Consider the flight of a tennis ball with mass \(57 \mathrm{~g}\) and a diameter of \(66.0\) mm. Assume the ball is served \(6.40\) meters from the net at a speed of \(50.0\) \(\mathrm{m} / \mathrm{s}\) down the center line from a height of \(2.8 \mathrm{~m}\). It needs to just clear the net \((0.914 \mathrm{~m})\).
1. Ignoring air resistance and spin, analytically find the path of the ball assuming it just clears the net. Determine the angle to clear the net and the time of flight.
2. Find the angle to clear the net assuming the tennis ball is given a topspin with \(\omega=50 \mathrm{rad} / \mathrm{s}\).
3. Repeat part b assuming the tennis ball is given a bottom spin with \(\omega=50 \mathrm{rad} / \mathrm{s}\).
4. Repeat parts \(a, b\), and \(c\) with a drag force, taking \(C_{D}=0.55\).
In Example \(3.7 a(t)\) was determined for a curved universe with nonrelativistic matter for \(\Omega_{0}>1 .\) Derive the parametric equations for \(\Omega_{0}<1\),

\[\begin{aligned} a &=\dfrac{\Omega_{0}}{2\left(1-\Omega_{0}\right)}(\cosh \eta-1) \\ t &=\dfrac{\Omega_{0}}{2 H_{0}(1-\Omega)^{3 / 2}}(\sinh \eta-\eta) \end{aligned} \label{3.67} \]

for \(\eta \geq 0\)

Find numerical solutions for other models of the universe.
1. A flat universe with nonrelativistic matter only with \(\Omega_{m, 0}=1\).
2. A curved universe with radiation only with curvature of different types.
3. A flat universe with nonrelativistic matter and radiation with several values of \(\Omega_{m, 0}\) and \(\Omega_{r, 0}+\Omega_{m, 0}=1\).
4. Look up the current values of \(\Omega_{r, 0}, \Omega_{m, 0}, \Omega_{\Lambda, 0}\), and \(\kappa .\) Use these values to predict future values of \(a(t)\).
5. Investigate other types of universes of your choice, but different from the previous problems and examples.