Page 460 - Excel for Scientists and Engineers: Numerical Methods

APPENDIX 8 ANSWERS AND COMMENTS FOR PROBLEMS 437

3. Set up the spreadsheet with three columns: x, y, y'. I used the Rungel custom
function. The exact expression for y is given in the answer spreadsheet.

4. Set up the spreadsheet with five columns: t, x, y, x' y'. Plot x vs. y to visualize
the trajectory. I used Goal Seek to find the value oft that makes y = 0.

5. Make a copy of the spreadsheet of problem 4 and modify it (I used the
Rungel custom function). The projectile struck the ground at x = 3 1967 m.
Note that the velocity was identical to that when it left the muzzle.

6. It may be helpful to set up the problem using the Euler method first, without
air drag, and then modify the spreadsheet to include air drag. Set up the
spreadsheet with eight columns: t, x, y, x' yl, x", y" and v.
If you experiment with different angles, it appears that an angle of about 30"
gives the longest drive when air resistance is taken into account.
For calculations and interesting discussion on Mickey Mantle's "tape
measure home run" of 565 feet, hit at Griffith Stadium on April 17, 1953, see
Grant R. Fowles and George L. Cassiday, Analytical Mechanics, 7'h ed.,
Brooks Cole.

7. Excel's SIN function requires angles in radians. It may be helpful to solve
the problem using the Euler method first.

8. The problem requires using two Runge-Kutta or Euler calculations. It may be
helpful to solve the problem using the Euler method first.

10. I used the Runge3 custom function to calculate the concentrations of A and
B. Note that the exact expressions fail if [A] = [B]; thus I made [B] very
slightly greater than [A].

11. I used names for the rate constants kl, kz, k3 and k4, to make the formulas
clearer; I used the Runge3 custom function to calculate the concentrations of
A, B and C.

Chapter I1 ODES with Boundary Values

1. Set up the spreadsheet as in Figure 1 1-2. Use an initial value of zero for the
slope. Then use Goal Seek to get the value of the slope (changing cell) that
gives a value of zero for the deflection at the other end of the beam (target
cell). Maximum deflection: 0.6138 in.

2. Use procedure as in problem 1. Maximum deflection: 0.9353 in at 200 in.