Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

1. Engr/Math/Physics 25 Prob 4.12Tutorial Bruce Mayer, PE Registered Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. Ballistic Trajectory • Studied in Detail in PHYS4A • The Height, h, and Velocity, v, as a Fcn of time, t, Launch Speed, v0, & Launch Angle, A h A t thit

3. For This Problem Parametric Description • h ~< 15 m • Or Equivalently: h  15m • [h ~< 15 m] & [v ~> 36 m/s] • Or By DeMorgan’s Theorem: ~([h  15m] | [v  36 m/s]) • [h < 5 m] I [v > 35 m/s] 40 m/s 9.81 m/s2 h 30° t • Find TIMES for Three cases

4. 1st Step → PLOT it • Advice for Every Engineer and Applied Mathematician or Physicist: • Rule-1: When in Doubt PLOT IT! • Rule-2: If you don’t KNOW when to DOUBT, then PLOT EVERYTHING

5. The Plot Portion of the solution File The Plot Plan % Bruce Mayer, PE * 21Feb06 % ENGR25 * Problem 4-12 % file = Prob4_12_Ballistic_Trajectory.m % % % INPUT PARAMETERS SECTION A = 30*pi/180; % angle in radians v0 = 40 % original velocity in m/S g = 9.81 % Accel of gravity in m/sq-S % % %CALCULATION SECTION % calc landing time t_hit = 2*v0*sin(A)/g; % divide flite time into 100 equal intervals t = [0: t_hit/100: t_hit]; % calc Height & Velocity Vectors as fcn of t h = v0*t*sin(A) - 0.5*g*t.^2 v = sqrt(v0^2 - 2*v0*g*sin(A)*t + g^2*t.^2) % % plot h & v %% MUST locate H & S Labels on plot before script continues plot(t,h,t,v), xlabel('Time (s)'), ylabel('Height & Speed'), grid • Then the Plot • Analyses Follow

6. Analyze the Plots • Draw HORIZONTAL or VERTICAL Lines that Correspond to the Constraint Criteria • Where the Drawn-Lines Cross the Plotted-Curve(s) Defines the BREAK POINTS on the plots • Cast DOWN or ACROSS to determine Values for the Break-Points • See Next Slide

7. Case a. Break-Pts 0.98 3.1

8. Case b. v Limits 1.1 3.05

9. Case c. v Limits v Limits 1.49 2.58

10. Advice on Using WHILE Loops • When using Dynamically Terminated Loops be SURE to Understand the MEANING of the • The LAST SUCEESSFULentry into the Loop • The First Failure Which Terminates the Loop • Understanding First-Fail & Last-Success helps to avoid “Fence Post Errors”

11. Solution Game Plan • Calc t_hit • Plot & Analyze to determine approx. values for the times in question • DONE • Precisely Determine time-points • For all cases • Divide Flite-Time into 1000 intervals → time row-vector with 1001 elements • Calc 1001 element Row-Vectors h(t) & v(t)

12. Solution Game Plan cont. • Case-a • Use WHILE Loops to • Count k-UP (in time) while h(k) < 15m • Save every time ta_lo = h(k) • The first value to fail corresponds to the value of ta_lo for the Left-side Break-Point • Count m-DOWN (in time) while h(m) < 15m • Save every time ta_hi = h(m) • The first value to fail corresponds to the value of ta_hi for the Right-side Break-Point

13. Solution Game Plan cont. • Case-b → Same TACTICS as Case-a • Use WHILE Loops to • Count k-UP While h(k) < 15m OR v(k) > 36 m/s • Save every time tb_lo = h(k) OR v(k) • The LastSuccessful value of tb_lo is ONE index-unit LESS than the Left Break point → add 1 to Index • Find where [h<15 OR v>36] is NOT true • Count m-DOWN While h(k) < 15m OR v(k) > 36 m/s • Save every time tb_hi = h(m) OR v(m) • The LastSuccessful value of tb_hi is ONE index-unit MORE than the Right Break point → subtract 1 from index • Find where [h<15 OR v>36] is NOT true

14. Solution Game Plan cont. • Case-c → Same TACTICS as Case-b • Use WHILE Loops to • Count k-UP while h(k) < 5m OR v(k) > 35 m/s • Save every time tc_lo = h(k) OR v(k) • The LastSuccessful value of tc_lo IS the Left-side Break-Point as the logical matches the criteria • Count m-DOWN while h(m) < 5m OR v(m) > 35 m/s • Save every time tc_hi = h(m) OR v(k) • The LastSuccessful value of tc_hi IS the Right-side Break-Point as the logical matches the criteria

15. Solution Game Plan cont. • MUST Properly LABEL the OutPut using the Just Calculated BREAK-Pts • Recall from the Analytical PLOTS • Case-a is ONE interval (ConJoint Soln) • ta_lo → ta_hi • Case-b is ONE interval (ConJoint Soln) • tb_lo → tb_hi • Case-c is TWO intervals (DisJoint Soln) • 0 → tc_lo • tc_hi → t_hit

16. Alternate Soln → FIND • Use FIND command along with a LOGICAL test to locate the INDICES of h associated with the Break Points • LOWEST index is the Left-Break • HIGHEST Index is the Right-Break • Same Tactics for 3 Sets of BreakPts • Again, MUST label Properly • Must INcrement or DEcrement “found” indices to match logical criteria • Need depends on Logical Expression Used

17. Compare: WHILE vs FIND • Examine Script files • Prob4_12_Ballistic_Trajectory_by_WHILE_1209.m • Prob4_12_Ballistic_Trajectory_by_FIND_1209.m • FIND is Definitely More COMPACT (fewer KeyStrokes) • WHILE-Counter is More INTUITIVE → Better for someone who does not think in “Array Indices”

18. Compare: WHILE vs FIND • While vs Find; Which is Best? • The “best” one is the one that WORKS in the SHORTEST amount of YOUR TOTAL-TIME

19. Portion of The m-file

20. Following are for projection onto Whiteboard

23. Portion of Plotm-file