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_lec14_ppt_

# _lec14_ppt_ - Lecture 14 P112 Agenda for today Challenge...

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Unformatted text preview: Lecture 14 P112 Feb 21, 2007 Agenda for today Challenge topic: Projectile motion with friction How, in practice, to solve differential equations NOT ON ANY EXAMS (Maybe a homework question) Differential Equations An equation in which derivatives of a function appear as variables They will cross your path often! e.g.Oscillations,... Example: Population growth: the number of bunny's born each year is proportional to the number of mommy bunnies dN = bN(t) dt b is a constant. Problem: derive an expression for N solving differential equations by "Separation of Variables" Population Growth: dN(t) = bN(t) dt 1 dN dt = bdt N dt 1 dN dt = bdt N dt Nf Ni dN = b dt N ti Nf Ni ) = b(t f b(t f t i ) tf log( ti ) N f = N ie Harder Differential equation Projectile motion -- with air resistance Break into components: No "analytic" solution Dimensional Analysis From m,g, and D, how do you make a velocity? Terminal velocity From m,g, and D, how do you make a time? Characteristic time Dimensionless variables: Calculate Extract Dimensionless Equations of Motion Suppose at =0, wx=1 and wy=0 What is w when =0.1? Take "timestep" Spreadsheet tau 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 wx 1 0.9 0.81850153 0.74970822 0.6898427 0.63638753 0.58766664 0.54258783 0.5004667 0.46089731 0.4236551 0.38862512 0.35575082 0.32499973 0.29634189 0.26973799 0.24513407 0.2224606 0.20163408 0.18255993 0.16513585 0.14925505 0.13480917 0.1216907 0.10979478 0.09902069 0.08927274 0.08046092 0.07250116 0.06531553 0.05883208 0.05298471 0.04771291 0.04296146 0.03868006 0.034823 wy 0 -0.1 -0.19094461 -0.27489612 -0.35294519 -0.42559585 -0.49301285 -0.55519476 -0.61209495 -0.66369967 -0.71007029 -0.75135802 -0.78779969 -0.8197023 -0.84742256 -0.87134574 -0.89186669 -0.90937423 -0.9242396 -0.93680853 -0.94739663 -0.9562873 -0.96373161 -0.96994947 -0.97513174 -0.9794427 -0.98302289 -0.98599183 -0.98845068 -0.99048474 -0.99216566 -0.99355348 -0.99469836 -0.99564212 -0.99641955 -0.99705958 |w| 1 0.90553851 0.8404788 0.79851756 0.77488919 0.76558534 0.76708132 0.77630069 0.79064982 0.80803687 0.82685154 0.84591274 0.86439979 0.88178041 0.89774356 0.91214143 0.92494157 0.9361892 0.9459784 0.9544309 0.96168093 0.9678649 0.97311465 0.97755338 0.98129343 0.98443542 0.9870682 0.98926935 0.99110603 0.99263595 0.9939084 0.99496528 0.99584203 0.99656857 0.99717003 0.9976675 wx |w| 1 0.81498466 0.68793318 0.59865517 0.53455165 0.48720897 0.4507881 0.42121131 0.3956939 0.37242202 0.35029987 0.32874294 0.30751094 0.28657839 0.26603902 0.24603919 0.22673469 0.20826521 0.19074148 0.17424084 0.15880799 0.14445872 0.13118478 0.11895915 0.1077409 0.09747948 0.08811829 0.07959752 0.07185634 0.06483454 0.05847369 0.05271794 0.04751452 0.04281404 0.03857059 0.03474177 wy |w| 0 -0.09055385 -0.1604849 -0.21950938 -0.27349341 -0.32582994 -0.37818095 -0.43099808 -0.48395276 -0.53629381 -0.58712271 -0.63558332 -0.68097389 -0.72279743 -0.76076815 -0.79479055 -0.82492457 -0.85134633 -0.8743107 -0.89411901 -0.91109327 -0.92555691 -0.93782135 -0.94817738 -0.95689036 -0.96419809 -0.97031063 -0.97541149 -0.97965943 -0.98319076 -0.98612178 -0.98855121 -0.99056243 -0.99222564 -0.99359972 -0.99473394 Results Strobe (velocities) 0 0 -0.2 0.2 0.4 0.6 0.8 1 1.2 1 1.5 Time Series -0.4 0.5 wy -0.6 0 0 -0.5 0.5 1 1.5 2 2.5 3 3.5 4 wy wx -0.8 w -1 -1 -1.2 wx -1.5 tau Approximate Solution: Improves as is made smaller Numerical integration: Gives x and y Example trajectories M=1 kg Vx(t=0)=1 m/s Time between dots: 0.1 s Total time: 5 s D=0.1kg/m D=0 D=0.01kg/m Another Example Mass on a spring: k x=0 M Push: Starts oscillating Dimensional Analysis: Characteristic time: Characteristic length: None (Scaling) Numerical Solution Suppose x(t=0)=10 cm=x0, v(t=0)=0 Dimensionless variables: Dimensionless equation of motion: "Initial conditions" Discretize Choose time step Smaller: more accurate, more time to calculate spreadsheet y 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 1.000 0.980 w 1.500 0.000 -0.100 -0.296 -0.480 Distance [x0] 1.000 0.500 0.000 0 -0.500 -1.000 -1.500 Time [t1] 5 10 15 20 25 30 35 40 45 0.921 0.825 -0.645 0.696 -0.784 0.539 -0.892 0.360 -0.964 0.168 -0.998 -0.032 -0.991 -0.230 -0.945 -0.419 Oscillation x=0 M k Restoring force pulls mass Overshoots Gets pulled back other way Summary Newton's laws: Differential equations Solving differential equations: Method of Finite Differences ...
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