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EEEEKKKKK!!!! In January, 1942 a Soviet Ilyushin 4 flown by - PowerPoint PPT Presentation

Dec 29, 2023 •366 likes •739 views

EEEEKKKKK!!!! In January, 1942 a Soviet Ilyushin 4 flown by Lieutenant I.M.Chisov was badly damaged by German gunfire. At an altitude of 21,980 feet Lieutenant Chisov fell from the plane. Unfortunately, he did not have a parachute on when he

  1. EEEEKKKKK!!!! In January, 1942 a Soviet Ilyushin 4 flown by Lieutenant I.M.Chisov was badly damaged by German gunfire. At an altitude of 21,980 feet Lieutenant Chisov fell from the plane. Unfortunately, he did not have a parachute on when he fell. He landed on the slopes of a snow-covered ravine and slid to the bottom. He suffered a fractured pelvis and severe spinal damage, but lived. By 1974 he had become Lieutenant Colonel Chisov. How fast was Lieutenant Chisov moving when he hit the ravine? How long did his fall take? Poor Lieutenant Chisov – p. 1/2

  2. The Drag Force Resistive Force on Coffee Filters 0.18 F f | = 1 Resistive Force (N) | � 2 DρAv 2 0.16 / ndf / ndf χ χ 2 2 0.0001598 / 10 0.0001598 / 10 p0 p0 0 0 0 0 ± ± 0.14 p1 p1 0 0 0 0 D - drag coefficient (dimen- ± ± 0.12 p2 p2 0.01589 0.01589 0.0002145 0.0002145 ± ± sionless). 0.1 ρ - air density ( kg/m 3 ). 0.08 A - Cross sectional area ( m 2 ). 0.06 v - speed ( m/s ). 0.04 0.02 0 0 0.5 1 1.5 2 2.5 3 3.5 4 v (m/s) t 2007-12-13 17:51:17 Poor Lieutenant Chisov – p. 2/2

  3. The Drag Force Aerodynamic forces acting on an artillery shell. The force � W is the drag or air resistive force, � L a is the lift, � F g is gravity, and the point D is the center of pressure. Note the change in the air resistive force at the speed of sound. Poor Lieutenant Chisov – p. 3/2

  4. Resistive Force on a Baseball Jonathan Papelbon gives Derek Jeter some chin music at 90 mph (40 m/s). What is the drag coefficient of a baseball? What is the resistive force at that speed? The mass of a baseball is 0.145 kg, its radius is 3.7 cm, and its terminal velocity is measured to be 43 m/s. Poor Lieutenant Chisov – p. 4/2

  5. Approximating a Function The plot below shows an arbitrary curve (black) with the tangent curve (red) at one point. 400 300 200 y � f � x � 100 0 a x 0 1 2 3 4 5 x Poor Lieutenant Chisov – p. 5/2

  6. Taylor Series for the Sine function The plot below shows the sine function. What are the first two nonzero terms of the Taylor series for the sine function expanded about the point θ = 0 ? How close do they come to the sine function? ) θ sin( 1 0.5 0 -0.5 -1 0 0 5 5 10 10 15 15 20 20 25 25 30 30 θ θ (radians) (radians) Poor Lieutenant Chisov – p. 6/2

  7. Taylor Series for the Sine function ) θ Sine function sin( 1 0.5 0 -0.5 -1 Taylor Series of order 3 0 0 5 5 10 10 15 15 20 20 25 25 30 30 (radians) (radians) θ θ ) θ Sine function sin( 1 0.5 0 -0.5 Taylor Series of Third Order -1 0 0 0.5 0.5 1 1 1.5 1.5 2 2 2.5 2.5 3 3 θ θ (radians) (radians) Poor Lieutenant Chisov – p. 7/2

  8. The Integral Convergence Test n 1 m versus ln � x � � ����� m � 1 8 6 f � x � 4 2 0 0 200 400 600 800 1000 1200 x � n Poor Lieutenant Chisov – p. 8/2

  9. Numerical Differentiation of a Curve A Computer Curve f 0 f 1 f −1 f(x) h f −2 f h 2 f −3 f 3 x x x x x x x −3 −2 −1 0 1 2 3 x Poor Lieutenant Chisov – p. 9/2

  10. Numerical Differentiation of a Curve A Computer Curve f 0 f 1 f −1 f(x) h f −2 f h 2 f −3 f 3 x x x x x x x −3 −2 −1 0 1 2 3 x Poor Lieutenant Chisov – p. 10/2

  11. Numerical Differentiation of a Curve A Computer Curve f 0 f 1 f −1 f(x) h f −2 f h 2 f −3 f 3 x x x x x x x −3 −2 −1 0 1 2 3 x Poor Lieutenant Chisov – p. 11/2

  12. Life is a Differential Equation (DE) Some definitions first Ordinary DE One independent variable; only total derivatives. Poor Lieutenant Chisov – p. 12/2

  13. Life is a Differential Equation (DE) Some definitions first Ordinary DE One independent variable; only total derivatives. Partial DE More than one variable so use partial derivatives. Poor Lieutenant Chisov – p. 12/2

  14. Life is a Differential Equation (DE) Some definitions first Ordinary DE One independent variable; only total derivatives. Partial DE More than one variable so use partial derivatives. Order of the DE Highest derivative in the DE. Poor Lieutenant Chisov – p. 12/2

  15. Life is a Differential Equation (DE) Some definitions first Ordinary DE One independent variable; only total derivatives. Partial DE More than one variable so use partial derivatives. Order of the DE Highest derivative in the DE. Degree of the DE Power of the highest order derivative in the DE. Poor Lieutenant Chisov – p. 12/2

  16. Life is a Differential Equation (DE) Some definitions first Ordinary DE One independent variable; only total derivatives. Partial DE More than one variable so use partial derivatives. Order of the DE Highest derivative in the DE. Degree of the DE Power of the highest order derivative in the DE. Linear DE No multiplications among dependent variables and their derivatives. Poor Lieutenant Chisov – p. 12/2

  17. Life is a Differential Equation (DE) Some definitions first Ordinary DE One independent variable; only total derivatives. Partial DE More than one variable so use partial derivatives. Order of the DE Highest derivative in the DE. Degree of the DE Power of the highest order derivative in the DE. Linear DE No multiplications among dependent variables and their derivatives. Homogeneous Every single term contains the dependent variables or their derivatives. Poor Lieutenant Chisov – p. 12/2

  18. Life is a Differential Equation (DE) Some definitions first Ordinary DE One independent variable; only total derivatives. Partial DE More than one variable so use partial derivatives. Order of the DE Highest derivative in the DE. Degree of the DE Power of the highest order derivative in the DE. Linear DE No multiplications among dependent variables and their derivatives. Homogeneous Every single term contains the dependent variables or their derivatives. Results of integrating the DE; n th order DE require General Solution n conditions to fix all the constants. Poor Lieutenant Chisov – p. 12/2

  19. Life is a Differential Equation (DE) Some definitions first Ordinary DE One independent variable; only total derivatives. Partial DE More than one variable so use partial derivatives. Order of the DE Highest derivative in the DE. Degree of the DE Power of the highest order derivative in the DE. Linear DE No multiplications among dependent variables and their derivatives. Homogeneous Every single term contains the dependent variables or their derivatives. Results of integrating the DE; n th order DE require General Solution n conditions to fix all the constants. General solution plus the n conditions. Particular Solution Poor Lieutenant Chisov – p. 12/2

  20. Life is a Differential Equation (DE) Some definitions first Ordinary DE One independent variable; only total derivatives. Partial DE More than one variable so use partial derivatives. Order of the DE Highest derivative in the DE. Degree of the DE Power of the highest order derivative in the DE. Linear DE No multiplications among dependent variables and their derivatives. Homogeneous Every single term contains the dependent variables or their derivatives. Results of integrating the DE; n th order DE require General Solution n conditions to fix all the constants. General solution plus the n conditions. Particular Solution Initial Value Problem Find v ( t ) and y ( t ) when v ( t 0 ) = v 0 or y ( t 0 ) = y 0 . Poor Lieutenant Chisov – p. 12/2

  21. Solving First-Order, Ordinary DEs 1. The differential equation and the initial conditions are given. For example dy dt = f ( t, y ) and y ( t 0 ) = y 0 and y ( t 1 ) is unknown. 2. Divide the range [ t 0 , t 1 ] into pieces. 3. Generate a recursion relationship between adjacent points. 4. Perform a step-by-step integration. Poor Lieutenant Chisov – p. 13/2

  22. Nuclear Decay The rate of radioactive decay of atomic nuclei is proportional to the number of nuclei N in the sample so dN dt = − λN where λ is a constant of proportionality (related to the half-life) and the negative sign means the number of nuclei is decreasing. The initial condition is N ( t = 0) = N 0 = 1000 . 1. Write down the analytical solution. 2. Generate an algorithm to solve this differential equation and apply it for the first three values of N ( t ) ‘by hand’ for h = ∆ t = 0 . 1 s and λ = 0 . 2 s − 1 . 3. Find the solution for the first sixty seconds. Poor Lieutenant Chisov – p. 14/2

  23. Nuclear Decay Results t(s) Calculation Result Analytic Result Poor Lieutenant Chisov – p. 15/2

  24. Solution of dN dt = − λN using an Euler algorithm (* initial parameter values *) Results N0 = 1000.0; t (s) N lambda = 0.2; t0 = 0.0; 0.0 1000.0 0.1 980.0 t1 = 60.0; 0.2 960.4 h = 0.1; 0.3 941.2 (* starting point of recursion relationship. *) Nn = N0; N 1000 (* make the table *) 800 table1 = Table[ 600 {t, 400 Nplus = Nn*(1 - lambda*h); 200 Nn = Nplus 60 t � s � }, 0 0 10 20 30 40 50 {t, t0 + h, t1, h}]; (* stick the starting point at the front of the table. *) table1 = Prepend[table1, {t0, N0}]; Poor Lieutenant Chisov – p. 16/2

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