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Engr/Math/Physics 25. Prob 4.12 Tutorial. Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. Ballistic Trajectory. Studied in Detail in PHYS4A The Height, h, and Velocity, v, as a Fcn of time, t, Launch Speed, v 0 , & Launch Angle, A. h. A. t. t hit.

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slide1

Engr/Math/Physics 25

Prob 4.12Tutorial

Bruce Mayer, PE

Registered Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

ballistic trajectory
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

parametric description
For This ProblemParametric 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
1 st step plot it
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
the plot plan
The Plot Portion of the solution FileThe 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
analyze the plots
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
slide7

Case a.

Break-Pts

0.98

3.1

slide8

Case b.

v Limits

1.1

3.05

slide9

Case c.

v Limits

v Limits

1.49

2.58

advice on using while loops
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”
solution game plan
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)
solution game plan cont
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
solution game plan cont1
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
solution game plan cont2
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
solution game plan cont3
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
alternate soln find
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
compare while vs find
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”
compare while vs find1
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