COMPUTER PROGRAMMING

Transcription

1 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 1 / 36 COMPUTER PROGRAMMING References: WITH FORTRAN Lecture Notes prepared by Prof. Dr. Namık Kemal ÖZTORUN Dr. Faruk TOKDEMİR, Programming with FORTRAN77, Middle East Technical University, Ankara, Microsoft Fortran Power Station Lahey Fortran [-1-]

2 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 2 / 36 CHAPTER 2 INTRODUCTION TO COMPUTER PROGRAMMING 2.1 An approach to problem solution Computers are utilized in solving a wide range of problems which in science, engineering, business, management and administration. Whatever the problem is, the computer cannot solve it by itself alone. It is only a tool, an electronic device that will accept information, process it, and give the results. It is a manmade machine that performs computations according to the commands or instructions given by humans. To get a solution to a problem from a computer, one should take the following steps. Problem Definition: This is a precise, well-defined, clearly understood set of statements of the problem. Problem analysis and the formulation of a method of solution: At this step, organization of input data, mathematical formulation and procedure for solution and organization of output are required. The method of solution is then expressed as a sequence of steps. Such a description of the method of solution is called an ALGORITHM. This is a verbal description of the solution in everyday language. The simple steps of solution expressed in everyday language are not a computer program and are not understood by a computer. It just aids to write a correct computer program later. Once the method of solution is formulated, then the logical steps and sequence of operation to be performed at each step are represented in a diagrammatic form. This is called a FLOWCHART. This is again not a computer program and not understood [-2-]

3 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 3 / 36 by a computer. A flowchart is a diagrammatic representation of the method of solution with special symbols. It is another way of describing and expressing an algorithm. By doing this, one can easily follow the flow of solution. After describing the method of solution by the algorithm and the flowchart, one can easily write the computer program by using a computer language. There are many computer languages. FORTRAN is one of them. Expressing the method of solution in terms of a computer language is called PROGRAMMING. The last step is to run the computer program on a computer to get the desired results. This step is repeated until all the errors are removed. This part includes testing and debugging. Debugging is simply searching and removing the errors discovered during running and testing Algorithmic approach and flowcharting As was defined earlier, the algorithm is a step by step procedure to solve a problem and the flowchart is the diagrammatic form of the algorithm. To illustrate these two concepts, let us start with the following problem. [-3-]

4 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 4 / 36 Operation A rectangle denotes a computation or a mathematical operation. It is also called an assignment box (A B + C) and has one exit. Other operation Decision A diamond is used to indicate a test, a decision. It is also called a decision box. If the decision is of type logical, than it has two exists for true and false. If the decision is of type arithmetic, than it has three exists for the signs (-, 0, +). In this case, the sign of the expression is tested. Data Predefined operation Internal storage Result This resembles a page torn off from a line printer. Output is shown on this figure. It has one exit. Results Start - Stop An oval shaped symbol indicates the beginning or end of a flowchart. Preparation Input from a keyboard. It has only one exit. Figure 2.1-a: The flowchart symbols [-4-]

5 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 5 / 36 Manual operation Connective A circle is a connection symbol and connects parts of a flowchart. Two circles carrying the same number or letter are considered to be connected by an arrow. Out-of-page connector Card A punch card figure indicates the input information (data) to the computer and has only one exit. Tape Toplam birleşimi Or Collate Harmanla Ranking Sıralama Extract Ayıkla Figure 2.1-b: The flowchart symbols [-5-]

6 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 6 / 36 Combine Birleştir Stored data Depolanmış veri Delay Gecikme Sequential access storage Sıralı erişimli depolama Magnetic disc Manyetik disk Direct access storage Doğrudan erişimli depolama Visualization Output to a terminal screen. It has one exit. Flow direction A flow symbol connects the symbols of the flowchart and indicates the direction of the flow. Figure 2.1-c: The flowchart symbols [-6-]

7 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 7 / 36 Example 2.1: How does a teacher give lecture? Suppose that the lecture hour starts at 10:40 and ends at 11:30. Solution to such a problem can take the following steps: Algorithm S1: Collect the notes in the office. S2: If the time is 10:30, lock the door of the office and leave. If not, wait. S3: Walk to the classroom. S4: Check the chalk, Is it enough? If it is not, find some more chalk. S5: If it is 10:40, start lecturing. If not, wait. S6: Check the time. If it is not 11:25 yet, continue lecturing. S7: If it is 11:25, then give exercises. S8: Check the time. Is it 11:30? If not, then ask if there are any question. Otherwise, next step. S9: Stop [-7-]

8 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 8 / 36 START B1 COLLECT NOTES B2, S1 B3, S2 IS TIME 10:30? NO WAIT YES WALK TO CLASSROOM B4, S3 1 B5, S4 IS THERE ENOUGH CHALK? NO FIND SOME YES 1 B6, S5 IS TIME 10:40? NO WAIT YES LECTURE A Flowchart box number: B, Algorithm statement number: S Figure 2.2-a: The flowchart of example 2.1 [-8-]

9 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 9 / 36 A CHECK TIME B7, S6 B8, S7 IS TIME 11:25? YES NO CONTINUE GIVE EXERCISE B9, S8 IS TIME 11:30? YES NO ANY QUESTION? STOP B10, S9 Flowchart box number: B, Algorithm statement number: S Figure 2.2-b: The flowchart of example 2.1 [-9-]

10 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 10 / : Sample algorithm and flowcharts The best way of learning how the solve problems using computers is to do practice. The following examples illustrate the method of solution using computers. Example 2.2: Write an algorithm and draw a flowchart to add 25 numbers. The numbers are to be input one at a time. Algorithm S1: Initially set the value of summation and incrementation to zero. S2: Input a number. S3: Add the input number to the value of summation. S4: Increment the value of counter by one. S5: Check the value of counter by 25. If it not 25, then go to step 2. S6: If it is 25, then output the value of summation. S7: Stop the work. The algorithm is completed. Or in more professional form; Use S as a name for summation Use X as a name for input numbers Use I as a name for incrementation Algorithm S1: S 0, I 0 S2: Input X S3: S S + X S4: I I + 1 S5: If I < 25, then go to S2 S6: Output S S7: Stop [-10-]

11 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 11 / 36 Flowchart Box number Algorithm Statement number FLOWCHART 1 START B1 2 S1 S 0 B2, S1 3 S1 I 0 B3, S1 4 S2 X B4, S2 5 S3 S S + X B5, S3 6 S4 I I + 1 B6, S4 7 S5 B7, S5 I < 25? TRUE FALSE 8 S6 SUM IS S B8, S6 9 S7 STOP B9, S7 Figure 2.3: Addition of tvetyfive numbers (Example 2.2) [-11-]

12 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 12 / 36 Example 2.3: Write an algorithm and draw a flowchart to find the mean (average) of N numbers. The value of N and the following N numbers are given each on a separate input record. Algorithm S1: Input N S2: S 0, I 0 S3: Input X S4: S S + X S5: I I + 1 S6: If I < N, then go to S3 S7: M S / N S8: Output the mean is M S9: Stop Trace for N=3 and X values: 4, 3, 5 N X I S M [-12-]

13 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 13 / 36 Flowchart Box number Algorithm Statement number FLOWCHART 1 START B1 2 S1 N B2, S1 3 S2 S 0 I 0 B3, S2 4 S3 X B4, S3 5 S4 S S + X B5, S4 6 S5 I I + 1 B6, S5 7 S6 B7, S6 I < N? TRUE FALSE 8 S7 M S / N B8, S7 9 S8 SUM IS S B9, S8 10 S9 STOP B10, S9 Figure 2.6: Mean of N numbers (Example 2.3) [-13-]

14 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 14 / 36 Example 2.4: Write an algorithm and draw a flowchart to compute the function F below for an unknown number of set of data. Each number is given on a separate input record. The last data record has the value 555 and defines the end of data. This is called a trailer card (line). Do not include this in calculation. F = X 2 1 if X 0 F = X if 0 < X < 1 F = X if X 1 Algorithm S1: Input X S2: If X = 555 then go to S8 S3: If X 0 then F X 2 1 S4: If 0 < X < 1 then F X S5: If X 1 then F X S6: Output X is X, F is F S7: Go to S1 S8: Stop Trace for the numbers: 0.25, -4, 16, 555 X F [-14-]

15 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 15 / 36 START B1 X B2, S1 X = 555? TRUE STOP B10 FALSE B3, S2 B7 X 0? TRUE F X 2 1 FALSE B4, S3 B8 0 < X < 1? B5, S4 TRUE F X FALSE F X B6, S5 X, F B9, S6 Flowchart box number: B, Algorithm statement number: S Figure 2.7: Computation of the function F (Example 2.4) [-15-]

16 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 16 / 36 Example 2.5: Write an algorithm and draw a flowchart to find the Greatest Common Divisor (GCD) of two integers. Algorithm S1: Input I, J S2: If I > J then interchange I and J S3: If I 0 then go to S6 S4: J J - I S5: Go to S2 S6: Print GCD is J S7: Stop Trace for I=12, J=15 Algebraically 12/2=6 [2] 15/3=5 [3] 6/2=3 [3] 5/5=1 [5] 3/3=1 [1] 1 1 I J = = = = =0 0 3 [-16-]

17 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 17 / 36 START B1 I, J B2, S1 S5 J J - I B3, S2 I > J F T B5, S4 K I I J J K B6, S4 F I 0 T B4, S3 GCD IS J B7, S6 STOP B8, S7 Flowchart box number: B, Algorithm statement number: S Figure 2.8: Greatest common divisor problem [-17-]

18 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 18 / 36 Example 2.6: The flowchart in Figure 2.10 gives a nontrivial solution to find the square root of a number within a given error by the Newton-Rapson method. The number and the error are given as input. There is no new concept in this solution. However, tracing this algorithm will add more in understanding how a certain algorithm or a flowchart works. The algebraic notation in box 15 is the absolute value of A-S 2. Use a calculator if possible. Tracing for A=2.5, ERR=0.01 A ERR X S A-S [-18-]

19 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 19 / 36 START B1 A, ERR B2 B3 A 0 T F B8 NO SQUARE ROOT EXISTS X 1 B4 S X (A / X) 2 B5 B6 A-S 2 >ERR T F THE SQUARE ROOT IS S B9 X S B7 B10 STOP Flowchart box number: B Figure 2.10: Square root of a number (Example 2.6) [-19-]

20 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 20 / 36 Example 2.7: Write an algorithm and draw a flowchart to find the smallest of N numbers. N and the numbers are given each on a separate input record. Suppose that the numbers are all less than Algorithm S1: ICOUNT 0, MIN 9999 S2: Input N S3: Input NUMBER S4: Increment ICOUNT by 1 (ICOUNT ICOUNT + 1) S5: If NUMBER < MIN then then MIN NUMBER S6: If ICOUNT < N then go to S3 S7: Output MIN S8: Stop The method of solution of this example finds the smallest of N numbers. The variable N determines the count of numbers and is given on the first record. The variable ICOUNT is used for incrementation and counts the numbers in process (box 14) When ICOUNT becomes equal to N, then the N numbers have been processed (box 17) and the smallest in MIN has been obtained. Initially, MIN is set to a greatest value (9999) because the problem says that all the numbers in the data set are assumed to be less than this number. If the problem was that on finding the largest of N numbers, then the initial value for that variable would be a number than the smallest of N numbers in the data set. Trace for N=5, and the numbers: -25, 43, -40, 0, 15 N NUMBER ICOUNT MIN [-20-]

21 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 21 / 36 START B1 ICOUNT 1 MIN 9999 B2, S1 N B3, S2 NUMBER B4, S3 ICOUNT ICOUNT+1 B5, S4 B6, S5 NUMBER < MIN T MIN NUMBER B7, S5 F B8, S6 T ICOUNT < N F MIN B9, S7 STOP B10, S8 Flowchart box number: B Figure 2.11: Smalest of N numbers (Example 2.7) [-21-]

22 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 22 / 36 Example 2.8: Write an algorithm and draw a flowchart to find the sum of even numbers from 2 to 100 (inclusive) Algorithm S1: ESUM 0, ENUM 2 S2: ESUM ESUM + ENUM S3: ENUM ENUM + 2 S6: If ENUM 100 then go to S2 S7: Output ESUM S8: Stop In the solution of this example, no input data is necessary. The necessary input data, which are even numbers from 2 to 100 (i.e., 2, 4, 6,.., 100), are generated in the algorithm. The variable ESUM keeps the sum of even numbers. The variable ENUM is used to generate the even numbers as well as incrementation. The result in ESUM is printed in box 600 in Figure [-22-]

23 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 23 / 36 START B1 ENUM 2 ESUM 2 B2, S1 ESUM ESUM + ENUM B3, S2 ENUM ENUM + 2 B4, S3 F ENUM 100 B5, S4 T ESUM B6, S5 STOP B7, S6 Flowchart box number: B, Algorithm statement number: S Figure 2.12: Sum of even numbers (Example 2.8) [-23-]

24 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 24 / 36 Example 2.9: Write an algorithm and draw a flowchart to find the sum of the first 100 terms of the series expansion given below considering each term separately for five different x values. The x values are given in degrees. 3 5 X X X X X X f (X) X 3! 5! 7! 9! 11! 13! Do not use the general term of the series expansion = Function Sine Algorithm S1 XCOUNT 1 S2 Input X S3 SIGN -1, TCOUNT 0, POWER 1, FACTOR 1 X X, TSUM X 180 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 POWER POWER+2 TCOUNT TCOUNT+1 XPOWER X POWER FACTOR FACTOR*(POWER-1)*POWER TERM XPOWER/FACTOR TSUM TSUM+SIGN*TERM SIGN -SIGN If TCOUNT<20-1 then go to S4 Output X, TSUM XCOUNT XCOUNT+1 If XCOUNT < 5 then go to S2 Stop Note: The series expansion is the trigonometric function SINE. For non-science students, this problem may seem to be highly mathematical. But the science students can easily understand that this in nothing but a series expansion of the trigonometric function sine. Whatever the function is, one can easily find a solution if he can observe its following properties. [-24-]

25 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 25 / 36 The sign of the terms is alternating. X values must be converted to the radian measure. This is a mathematical requirement. The power of X are odd integers. The factorial in the denominator is just the factorial of an odd power of a term. For example, in the second term, the power of X is 3 and 3! = = 6 in its denominator. Then, investigate a term which will be a base for the method of solution. In this representation, the variable N is used for the odd numbers of the powers of terms; the variable XPOWER represents X N ; the variable FACTOR is the factorial of odd powers; the variable SIGN will keep the alternating sign of the terms; the variable TERM is just a term in the series; TERM = X N /FACTOR and TCOUNT counts the terms included. For example, for the following term of the function; X 3 3! SIGN = POWER = 3 XPOWER = 3 X FACTOR = 3! TERM = X 3 3! Introducing XCOUNT for counting the number of X values, and X for imputing numbers, we can write the following algorithm showing our method of solution taking the first 20 terms of the series expansion expression for 5 different X values [-25-]

26 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 26 / 36 START B0 XCOUNT 1 B1, S1 X B2, S2 SGN -1 TCOUNT 0 POWER 1 FACTOR 1 X X 180 TSUM X B3, S3 T STOP F XCOUNT 5 B15, S3 B14, S3 POWER POWER + 2 TCOUNT TCOUNT + 1 XPOWER X POWER B4, S4 B5, S5 B6, S6 B13, S3 XCOUNT XCOUNT + 1 FACTOR FACTOR (POWER - 1) POWER TERM XPOWER / FACTOR TSUM TSUM + SGN TERM B7, S6 B8, S6 B9, S6 T SGN -SGN B10, S6 B12, S6 ICOUNT 20-1 F X, TSUM B11, S6 Flowchart box number: B, Algorithm statement number: S Figure 2.13: A flowchart to compute the series expansion given in Example 2.9 [-26-]

27 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 27 / Arrays in Algorithms Example 2.10: Write an algorithm and draw a flowchart to find the largest of N numbers. N and the numbers are given on separate records. Use a subscripted variable for inputting numbers. (See also Example 2.7) Algorithm S1: Input N S2: Input X i {i from 1 to N in steps of 1} or, X i {I=1(1)N} S3: L X 1 S4: i 2 S5: If L < X i then L X İ S6: i i+1 S7: If i N then go to S5 S8: Output Largest value is L S9: Stop A solution to this problem will bring about a new concept. It is the use of subscripted variables in the method of solution. A subscripted variable X with its index number i will define which number is in process. Since we have N numbers, the subscript i will vary from 1 to N. For example X 1, X 2,., X N are used for the 1st, 2nd,., Nth numbers respectively. The subscripted variable name X is known as an array of one dimension or a vector and has N elements. Follow the method of solution below and try to understand how subscripted variable names or array elements are handled. The numbers are read in the subscripted variable X i whose subscript I varies from 1 to N in steps of one in box 3 in Figure Now, all the numbers are ready for processing so that no additional input for the numbers are needed. The initial value for the largest number which is shown by L is set to the first number X 1 in box 4 and the search in the loop starts with X 2. Later in the loop initial value is replaced by a large value if the condition in the decision box 6 is true. The loop is controlled by the boxes 8 and 9. On false exit from box 9, we proceed the box 10 where we output the largest number present in X i s. Tracing using the sample data illustrates how the method of solution works. [-27-]

28 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 28 / 36 Trace using the data N=5, and the numbers: 3, 2.5, 5, 1, 3.5 i X L [-28-]

29 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 29 / 36 START B1 N B2, S1 X i [ i = 1 (1) N ] B3, S2 L X 1 B4, S3 i 2 B5, S4 B6, S5 L < X i T L X i B7 F i i + 1 B8, S6 T i N B9, S7 F B11, S9 B10, S8 LARGEST IS L STOP Flowchart box number: B, Algorithm statement number: S Figure 2.14: Largest of N numbers in an array (Example 2.10) [-29-]

30 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 30 / 36 Example 2.11: Write an algorithm and draw a flowchart to input numbers into two subscripted variables X i and Y i and then find the addition of their elements as the elements of a new subscripted variable Z i. Output the resultant subscripted variable Z i. The input records are arranged so that they should be read in as X 1, Y 1, X 2, Y 2,, X 100, Y 100 Algorithm S1: Input (X i, Y i,){i=1 to 100 in step of 1} or, (X i, Y i ){i=1(1)100) S2: i 1 S3: Z i X i +Y i S4: i i+1 S5: If i 100 then go to S3 S6: Output Z i {i=1(1)100} S7: Stop Trace for the numbers: 5, 2.5, -3, 4, 2.5, 5, -2, 6 After reading: X i ={5, -3, 2.5, -2} and Y i ={2.5, 4, 5, 6} i X i Y i Z i [-30-]

31 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 31 / 36 START B1 ( X i, Y i ) [ i = 1 (1) 100 ] B2, S1 i 1 B3, S2 Z i X i + Y i B4, S3 i i + 1 B5, S4 F i 100 B6, S5 T Z i [ i = 1 (1) 100 ] B7, S6 STOP B8, S6 Flowchart box number: B, Algorithm statement number: S Figure 2.15: Addition of two subscripted variable names (Example 2.11) [-31-]

32 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 32 / 36 Example 2.12: Input nine numbers one at a time and locate each as an element of a doubly subscripted variable X ij. Each subscript has the range from 1 to 3. Output the resultant doubly subscripted array X ij. Suppose that the first index i runs faster than the second index j (i.e., for each value of the index j, the index i takes 1,2,3) A doubly subscripted variable is known to be a two dimensional array or a matrix. The first index defines the row and the second index defines the column numbers of a matrix. The last number of the rows and columns are the maximum values of corresponding indices. The Figure 2.16 displays a schematic representation of a two dimensional array or a matrix with three rows and three columns. [ X] X X X X X X X X X Figure 2.16: A 3x3 matrix Algorithm S1: i 1, j 1 S2: Input X ij S3: i i+1 S4: If i 3 then go to S2 S5: i 1, j j+1 S6: If j 3 then go to S2 S7: Output X i {i,j=1(1)100} S8: Stop [-32-]

33 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 33 / 36 Trace for the numbers: 5, 7, -8, 2, 1, 4, 5, 3, 6 i j X ij The matrix will be 5 [ X ij ] [-33-]

34 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 34 / 36 START B1 İ 1 J 1 B2, S1 X i j B3, S2 İ 1 B4, S3 T İ 3 B5, S4 F İ 1 J J + 1 B6, S5 T J 3 B7, S6 F X i j [ 1 (1) 3 ] B8, S7 STOP B9, S8 Flowchart box number: B, Algorithm statement number: S Figure 2.17: A flowchart for Example 2.12 [-34-]

35 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 35 / 36 Example 2.13: Write an algorithm and draw a flowchart to describe a projectile motion for T=0, 1, 2,..10 seconds if the initial speed V 0 at an angle θ with respect to horizontal is given. Output T, V X, V Y, V. To solve this physics problem, the necessary formulation for the associated velocities must be given or must be obtained from a reference book. If V 0 and θ in degrees are given, the subsequent projectile motion at a specific T second is described as VX 0 VY V0 V V Cos( ) Sin( ) g t 2 V X V 2 Y One can write the following solution using this information. Algorithm S1: Input V 0, ANGLE S2: T 0, ANGLE ANGLE * π /180 S3: V X V 0 * COS(ANGLE) V Y V 0 * SIN(ANGLE) G * T V 2 X V V 2 Y S4: Output T, V X, V Y, V S5: T T + 1 S6: If T 10 Then go to S3 S7: Stop [-35-]

36 Prof. Dr. Namık Kemal ÖZTORUN Lecture Notes for Computer Programming Course Page 36 / 36 START B1 V 0, ANGLE B2, S1 T 0 ANGLE ANGLE 3.142/ 180 B3, S2 V X V 0 COS(ANGLE) V Y V 0 SIN(ANGLE) G T V 2 X V V 2 Y B4, S3 T, V X, V Y, V B5, S4 T T + 1 B6, S5 T T 10 B7, S6 F STOP B8, S7 Flowchart box number: B, Algorithm statement number: S Figure 2.18: The projectile motion problem (Example 2.13) [-36-]