1. INTRODUCTION TO MATLAB
Creating basic scripts

2. PROJECTILE 1

3. POWEER!
Stupid pig..
Angry Birds®

4. Newton`s equation
wektory

5. Newton`s equation
Dv/dt = a!

6. Dv/dt=a

7. Newton`s equation – algorithmwe

8. Newton`s equation – algorithm
Nad r wektor

9. Newton`s equation – algorithm example

10. Newton`s equation – algorithm example

11. Projectile in LibreOffice Calc

12. t: 0 to 3 [s]

13. t: 0 to 3 [s]

14. Acceleration in x direction is 0.
RADIAN
X(t-1)
Acceleration in x direction is 0.

23. PLOT

24. scatter

26. Now, let`s try to change initial values of the velocity, Δt and initial angle.
What do you observe?

27. PROJECTILE TASK
CALCULATE AND PLOT REAL VALUE OF THE Y COORDINATE. COMPARE IT WITH THE VALUE OF Y CALCULATED WITH NEWTON`S EQUATION.

30. TO NAME YOUR SERIES FOR PLOT

34. PROJECTILE IN MATLAB– initial values
clc
clear all
v=100; % [m/s] initial value of velocity
alpha=pi/4; % [rad] initial value of angle
g=9.81; % [m/(s^2)] gravity constant
dt=0.1; %[s] single time step
In MATLAB you are not supposed to define type of variable

35. PROJECTILE – initial values
%position
x(1)=0; %[m]
y(1)=0; %[m]
% initial values of vx and vy
vx(1)=v*cos(alpha); %[m/s]
vy(1)=v*sin(alpha); %[m/s]
Number one in brackets mean, that x variable is list (contains many values), and now first value is being defined

36. PROJECTILE – initial values
%position
x(1)=0; %[m]
y(1)=0; %[m]
% initial values of vx and vy
vx(1)=v*cos(alpha);%[m/s]
vy(1)=v*sin(alpha);%[m/s]
Number one in brackets mean, that x variable is list (contains many values), and now first value is being defined

37. PROJECTILE - loop
for s=2: % for each s from 2 to 144. Started from 2, because 1 is initial value (already defined)
vx(s)=vx(s-1);
vy(s)=vy(s-1)-g*dt; %new vy = previous vy - gravity in delta t
x(s)=x(s-1)+vx(s)*dt; %new position
y(s)=y(s-1)+vy(s)*dt;
end
Loop starts with for (and condition), then repeatable intructions - each after indent, and 'end'.
New vx (in point s) = vx in previous point (s-1)

38. BUT SOME EXAMPLES FIRST
PROJECTILE - loop
BUT SOME EXAMPLES FIRST
for s=2: % for each s from 2 to 144. Started from 2, because 1 is initial value (already defined)
vx(s)=vx(s-1);
vy(s)=vy(s-1)-g*dt; %new vy = previous vy - gravity in delta t
x(s)=x(s-1)+vx(s)*dt; %new position
y(s)=y(s-1)+vy(s)*dt;
end
Loop starts with for (and condition), then repeatable intructions - each after indent, and 'end'.
New vx (in point s) = vx in previous point (s-1)

39. PROJECTILE – loop example
for s=2
vx(2)=vx(2-1);
vy(2)=vy(2-1)-g*dt; %new vy = previous vy - gravity * timestep
x(2)=x(2-1)+vx(2)*dt; %new position
y(2)=y(2-1)+vy(2)*dt;
end

40. PROJECTILE – loop example
for s=2
vx(2)=vx(2-1);
vy(2)=vy(2-1)-g*dt; %new vy = previous vy - gravity * timestep
x(2)=x(2-1)+vx(2)*dt; %new position
y(2)=y(2-1)+vy(2)*dt;
end

41. PROJECTILE – loop example
for s=3
vx(3)=vx(3-1);
vy(3)=vy(3-1)-g*dt; %new vy = previous vy - gravity * timestep
x(3)=x(3-1)+vx(3)*dt; %new position
y(3)=y(3-1)+vy(3)*dt;
end

42. PROJECTILE – loop example
for s=3
vx(3)=vx(3-1);
vy(3)=vy(3-1)-g*dt; %new vy = previous vy - gravity * timestep
x(3)=x(3-1)+vx(3)*dt; %new position
y(3)=y(3-1)+vy(3)*dt;
end

43. PROJECTILE - loop
for s=2: % for each s from 2 to 144. Started from 2, because 1 is initial value (already defined)
vx(s)=vx(s-1);
vy(s)=vy(s-1)-g*dt; %new vy = previous vy - gravity in delta t
x(s)=x(s-1)+vx(s)*dt; %new position
y(s)=y(s-1)+vy(s)*dt;
end
Loop starts with for (and condition), then repeatable instructions - each after indent, and 'end'.
New vx (in point s) = vx in previous point (s-1)

44. PROJECTILE - plot plot (x,y) xlabel('distance') ylabel('height')
Plot will be opened in new window while executing script
Plot(x-axis values list, y-axis values list)
Every text in code is required to be in quotes

45. PROJECTILE - results

46. CHANGE: TO RUN CALCULATION plot (x,y) xlabel('distance')
ylabel('height') TO
plot1(x,y,dt,vx,vy)
RUN CALCULATION

47. IF YOU PLOT ALL ENERGIES OVER TIME WHAT DO YOU OBSERVE?
PROJECTILE TASK
ADD TO THE LOOP AND CALCULATE POTENTIAL ENERGY, KINETIC ENERGY AND TOTAL ENERGY, THEN PLOT THEM.
IF YOU PLOT ALL ENERGIES OVER TIME WHAT DO YOU OBSERVE?
Masa!!!!!!!!!!!!!!!!!!!!!!

48. IF YOU PLOT THESE ENERGIES OVER TIME WHAT DO YOU OBSERVE?
PROJECTILE TASK
ADD TO THE LOOP AND CALCULATE POTENTIAL ENERGY, KINETIC ENERGY AND TOTAL ENERGY, THEN PLOT THEM.
IF YOU PLOT THESE ENERGIES OVER TIME WHAT DO YOU OBSERVE?
Remember to add the initial values of the energies before the
loop…

49. PROJECTILE – initial value of energy
%energy - initial values
ep(1)=g*y(1)*m; %potential energy
ek=(vx(1)^2+vy(1)^2)*m/2; %kinetic energy
ec(1)=ek(1)+ep(1); %total energy

50. PROJECTILE– energies added to loop
for s=2:144
vx(s)=vx(s-1);
vy(s)=vy(s-1)-g*dt;
x(s)=x(s-1)+vx(s)*dt;
y(s)=y(s-1)+vy(s)*dt;
ep(s)=g*y(s)*m;
ek(s)=(vx(s)^2+vy(s)^2)*m/2;
ec(s)=ep(s)+ek(s);
t(s)=t(s-1)+dt;
end

51. PROJECTILE - plot plot (t,ec) xlabel('time') ylabel('energy')
legend ('total energy','Location', 'NorthEast')
% plot of each energy
plot (t,ec,t,ek,t,ep)
legend ('total energy','kinetic energy','potential energy','Location', 'NorthEast')

52. Result!

53. NOW CHANGE VALUE OF STEPS TO 2:150
Result!
NOW CHANGE VALUE OF STEPS TO 2:150

54. PROJECTILE - results

55. PROJECTILE - results

56. SPRING
One mass

57. L0

58. L0 x

59. 1 2
Equilibrium position 5 3 s 4
Attachment point

61. SPRING – initial values
clc
clear all
%initial values
x=4; %extention
k=0.1; %spring constant
m=1; %mass
dt=0.1; % time step

62. SPRING – initial values
t(1)=0; %time
x(1)=x; %position
v(1)=0; %velocity

63. SPRING - loop for i=2:510 v(i)=v(i-1)+(-k*x(i-1)/m)*dt;
x(i)=x(i-1)+dt*v(i);
t(i)=t(i-1)+dt;
end

64. SPRING - plot plot(t,x) xlabel('time')
legend('position','Location','NorthEast')

65. SPRING - results

66. 1 2
Equilibrium position 5 3 s 4
Attachement point

67. RUN CALCULATION ADD ENERGIES (to the loop) THEN ANIMATION
ep(i)=k*x(i)^2/2;
ek(i)=m*v(i)^2/2;
ec=ep+ek;
THEN ANIMATION
plot3(x,dt,v,t,ep,ek,ec)
RUN CALCULATION
Kulka i wykres sinusoidy> energie rysowane> nie są zachowane bo dt jest małe > jak polepszyć??? > zadanie z kulką > feature 3
subplot(2,2,4)
xlim([-limits(6) limits(6)])
ylim([-limits(7) limits(7)])
xlabel('position (x)')
ylabel('velocity (v)')
grid
hold on
Do loop
subplot(2,2,4);
trace4=plot(y(k),v(k),'y:o',...
'MarkerFaceColor',[.0 k/numel(y) .7058],...
'Color',[.0 k/numel(y) .7058],...
'LineWidth',3.5);

68. ARE THE CALCULATIONS CORRECT?
What about the total energy?
ARE THE CALCULATIONS CORRECT?

69. ENERGY CONSERVATION LAWEp Ek

70. ENERGY CONSERVATION LAW

71. ENERGY CONSERVATION LAW
Looks familiar?

72. ENERGY CONSERVATION LAW
Looks familiar?

73. ENERGY CONSERVATION LAW
ADD after your loop:
v=v*sqrt(m/k);
PLOT:
plot(x,v)

74. RESULT A

75. CHANGE:
plot(x,v) TO
plot4(v,x,dt)
RUN CALCULATION

76. SPRING
Two masses

77. 1 2 L

78. 2 1 L
Δx1
L+Δx2

79. F1= k*Δl
F2= -k*Δl 1 2 x1 x2
-0.1 L
0.1
X2-x2=
Odwrocic dl

80. SPRING – initial values
clc
clear all
l=2;
m=1;
k=3;
dt=0.1;

81. SPRING – initial values
x1(1)=-0.5;
x2(1)=l+1;
t(1)=0;

82. SPRING - loop

83. SPRING - plot plot(t,x1,t,x2) xlabel('t') ylabel('x')
legend ('x1','x2','Location', 'NorthWest')

84. SPRING - results

85. CHANGE: TO RUN CALCULATION plot5(x1,x2,dt,v1,v2,t) plot(t,x1,t,x2)
xlabel('t')
ylabel('x')
legend ('x1','x2','Location', 'NorthWest') TO
plot5(x1,x2,dt,v1,v2,t)
RUN CALCULATION

86. IF YOU PLOT ALL ENERGIES OVER TIME WHAT DO YOU OBSERVE?
SPRING TASK
ADD TO THE LOOP AND CALCULATE POTENTIAL ENERGY, KINETIC ENERGY AND TOTAL ENERGY, THEN PLOT THEM.
IF YOU PLOT ALL ENERGIES OVER TIME WHAT DO YOU OBSERVE?

87. RESULTS

88. THE PURPOSE! Knowledge and experience = endless possibilieties
Creating models of particles,
such as chloride and hydrogen.
Working only with computer
and creating real experiment. cl

89. Cl2 molecule k = 322.7 N/m – spring constant for chlorine system
𝜇= 𝑚 1 𝑚 2 𝑚 1 + 𝑚 2 =2.903∗ 10 −26 𝑘𝑔 – reduced mass
x = 5*10-12 m – typical atomic displacement
Two body problem reduced to one body problem by introduction of reduced mass: cl
𝑑 2 𝑥 𝑑 𝑡 2 + k 𝜇 x=0

90. We can use our program to get…
Atomic motion
Spectrum
Fourier’s transform cl
wavenumber
1.678*1013
𝜈 = 𝑐 𝑚 −1
1/1.678*1013

91. SEE YOU TOMMOROW!