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

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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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Engr/Math/Physics 25

Prob 4.12Tutorial

Bruce Mayer, PE

Registered Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

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

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
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 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
• 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

Case a.

Break-Pts

0.98

3.1

Case b.

v Limits

1.1

3.05

Case c.

v Limits

v Limits

1.49

2.58

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