## Problems

Answers to the following problems are found in the folder "Ch. 10 (ODE)" in the "Problems & Solutions" folder on the CD.

1. A function is described by the differential equation dyjdt = 1 -t\[y . Calculate y for t = 0 to t =5, in increments of 0.1.

2. A function is described by the differential equation dy 2x2 /(I + x2) dx 1 + x2

3. A function is described by the differential equation dx y - arctan(x) h--

Calculate y for x = 0 to x = 2.5. Adjust the magnitude of Ax for different parts of the calculation, as appropriate.

4. Trajectory I. Consider the motion of a projectile that is fired from a cannon. The initial velocity of the projectile is v0 and the angle of elevation of the cannon is 6 degrees. If air resistance is neglected, the velocity component of the projectile in the x direction (x1) is v0 cos 9 and the component in the y direction is v0 sin 6-gt. Use Euler's method to calculate the trajectory of the projectile. For the calculation, assume that the projectile is a shell from a 122-mm field howitzer, for which the muzzle velocity is 560 m/s. (Getting started: create five columns, as follows: t, x', y\ x, y. Calculate x and y, the coordinates of distance traveled, from, e.g., xi+\ = x, + x/At.) Verify that the maximum range attainable with a given muzzle velocity occurs when 6 = 45°.

5. Trajectory II. Without air resistance, the projectile should strike the earth with the same / that it had when it left the muzzle of the cannon. Because of accumulated errors when using the Euler method, you will find that this is not true. Repeat the calculation of problem number 1 using RK4.

6. Trajectory III. To produce a more accurate estimate of a trajectory, air drag should be taken into account. For speeds of objects such as baseballs or cannonballs, air drag can be taken to be proportional to the square of the velocity,/ = Dv2. The proportionality constant D - 0.5pCA, where p is the density of air, A is the cross-sectional area of the projectile and C, the drag coefficient, is a dimensionless quantity that depends on the shape of the projectile. The forces acting on a projectile in flight are illustrated in the following figure.

Combining the above equation for the air drag and the relationship between force and acceleration, / = ma, we get, for the "deceleration" in the redirection, x" ' -Dv2/m; y" = -Dv2/m-g.

Calculate the trajectory of a baseball hit at angle 0= 30° with initial velocity 50 m/s. The parameters of the baseball are: mass 145 g, circumference 23 cm (from Rules of Baseball, Major League Baseball Enterprises, 1998). For air resistance, use p = 1.2 kg/m2 and the drag coefficient C = 0.5. (Getting started: create eight columns, as follows: t, x", y", x\ y\ v, x, y. At t = 0, x' and y' are calculated as in the previous problem, but for subsequent t values, they are calculated by the Euler method, using the previous values of x" and y". Calculate * and y, the coordinates of distance traveled, using, e.g., xM =x, + x,'A t + Vtx"(M)2.)

7. Pendulum Motion I. The motion of a simple pendulum, consisting of a mass M at the end of a rod of length L, is described by the following firstorder differential equation:

where co = angular velocity (rad/s)

6= angle of displacement from equilibrium position g = 9.81 m/s2 1= 1.0 m

Calculate the angular velocity of the pendulum beginning with the initial conditions 6= 10°, a= 0.3.

dQ L co

8. Pendulum Motion II. The motion of a simple pendulum as a function of time is described by the following second-order differential equation:

dt2 L

where the terms in the equation are as defined in the preceding problem. Generate a table of angle of displacement as a function of time from t = 0 to t = 2 seconds, with 6= 10° and dd/dt = 0 at t = 0 .

9. Liquid Flow. A cylindrical tank of diameter D is filled with water to a height h. Water is allowed to flow out of the tank through a hole of diameter d in the bottom of the tank. The differential equation describing the height of water in the tank as a function of time is

where g is the acceleration due to gravity. Produce a plot of height of water in the tank as a function of time for D = 10 ft, d = 6 in and h0 = 30 ft.

Compare your results with the analytical solution h = (*J~h^ - kt/2) , where k -(d2 /D2)*j2g .

10. Chemical Kinetics I. Calculate concentrations as a function of time for the second-order reaction k

for which -d[A]/dt = -d[B]/dt = d[C]/dt = k[A][B], Use [A]0 = 0.02000, [B]o = 0.02000, k = 0.( from 0 to 500 seconds.

[B]o = 0.02000, k = 0.050 s"1. Calculate concentrations over the time range

11. Chemical Kinetics II. Use the Runge custom function to calculate [A], [B] and [C] for the coupled reaction scheme

using [A]0 = 0.1, [B]0 = 0, [C]0 = 0 mol L'\ k\ = 1 s"1, k2 = 1 s"', h = 0.1 s-1 and k4 = 0.01 s~', over the range 0-100 s.

12. Chemical Kinetics III. Repeat #8, using [A]0 = 0, [B]0 = 0.1, [C]0 = 0 mol L"1

13. Chemical Kinetics IV. Repeat #8, using [A]0 = 0, [B]0 = 0, [C]0 = 0.1 mol L-1.

0 0