System Design for Optimum Performance

 

Optimum Launch Angle of a Projectile for Maximum Distance

 

 

1.   Understand the purpose of the problem.

·       We will be putting an initial velocity of 100 feet/sec on a projectile known to be small and heavy.  We want to determine the launch angle that will give the projectile the greatest distance.  This is what we are calling the optimum launch angle in this application.

 

2.   Collect the known information. Realize that some of it might be discovered later to be unnecessary.

·       Initial velocity is 100 feet/sec.

·       We know the object is small and heavy and thus should not be slowed much by air resistance. 

3.   Determine what information you must find.

·       We must determine the distance the projectile will travel as a function of the launch angle and initial velocity.  We will then search the angles to see which gives us the most distance.

 

4.   Simplify the problem only enough to allow the required information to be obtained. State any assumptions you make.

·       We assume that air resistance is negligible.

·       We assume that the ground is flat over the range used (we are not shooting off a cliff).

 

5.   Draw a sketch and label any necessary variables.

 

 

 

 

 

 

 

 

 

 


6.   Determine which of the fundamental principles are applicable.

·       Component velocities found using trigonometric identities (SOH CAH TOA):

·       We can treat the vertical flight and the horizontal flight independently.

·       Vertical flight is given by (See Sample Problem 6-5 p. 155):

·       The roots of the vertical flight are the times when the object is on the ground.  Once at the beginning and once at the end.  The roots are t=0 and t=2vy/g.  The second is the length of the flight (time to hitting ground)

·       The horizontal trajectory is simply the initial horizontal velocity times time (assuming no air resistance):

·       The total distance traveled is

 

7.   Think about your proposed solution approach in general and consider other approaches before proceeding with the details.

·       Since we have a mathematical model for the system based on our initial assumptions, we could mathematically optimize using calculus.

·       We could solve the equations on a spreadsheet and search for the optimum value.

 

8.   Label each step of the solution process.

·       Compute the component velocities from angle and initial velocity

·       Determine time of flight tg

·       Distance traveled is vx times tg

·       Search over a range of angles for the one producing the maximum distance

 

9.   Solve the problem. If you are solving the problem with a program:

 

·       State the problem concisely.

Develop a MATLAB function, which determines X as a function of angle and initial velocity v using the equations above.  Find the angle yielding the maximum distance via a search.

 

·       Specify the data to be used by the program. This is the “input”.

v=100 feet/sec.

Reasonable angles are in the range 0 to 90 degrees.

 

·       Specify the information to be generated by the program. This is the “output”.

 

The distance X for each angle.

 

·       Work through the solution steps by hand or with a calculator; use a simpler data set if necessary.

 

Angle of 0 degrees or 90 degrees should yield 0 distance.

Must be nonzero in between.

 

·       Write and run the program.

 

function X=projdist(angd,v)

%

% X=projdist(angd,v)

%

% Computes the distance a projectile will fly across flat ground

% with no air resistance, given the initial velocity and angle of

% launch.

%

% X      -  output distance (same dimensions as angd) (feet)

% angd - input angle or array of angles (degrees)

% v       - input scalar initial velovity (feet/sec)

%

% Author: Dr. Russell C. Hardie

% Date: 1/15/01

 

g=32.17;  % acceleration due to gravity 32.17 ft/sec^2

 

% create x-y velocities for each angle entered

angr=deg2rad(angd);

vx=v*cos(angr);

vy=v*sin(angr);

 

% Find the time we hit ground

tg=2*vy/g;

 

% horizontal path (distance is velocity  x time)

X=vx.*tg; % total distance traveled

 

·       Check the output of the program with your hand solution.

 

>>          X=projdist(0,100)

X=0

 

>>          X=projdist(90,100)

X=3.8270e-014

 

>>          X=projdist(30,100)

X=270.6329

 

Hand calculated value checks out!

 

·       Run the program with your input data and perform a reality check on the output.

 

>>          A=[0:1:90];

>>          X=projdist(A,100);

>>          plot(A,X)

>>          xlabel('Angle (degrees) ');

>>          ylabel('Distance Traveled (feet) ');

>>          title('Projectile Analysis');

 

>>          [maxX,n]=max(X);

>>          A(n)

 

 

Looks OK?

 

 

·       If the program will be used as a general tool in the future, test it by running it for a range of reasonable data values; perform a reality check on the results.

 

 

10.                   Present your results.

 

Well, what did you observe?

 

Do not state the answer with any greater precision than is justified by any of the following:

·       The precision of the given information.

·       The simplifying assumptions.

·       The requirements of the problem.

 

The angles were only searched by integers, so the maximum precision of the optimum angle is to the nearest integer.  The maximum distance traveled should not be stated with an accuracy greater than that of the input velocity or g.