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by: Jalon Willms
Jalon Willms
GPA 3.69


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About this Document

Class Notes
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This 3 page Class Notes was uploaded by Jalon Willms on Monday October 19, 2015. The Class Notes belongs to PH 235 at Rose-Hulman Institute of Technology taught by Staff in Fall. Since its upload, it has received 10 views. For similar materials see /class/225118/ph-235-rose-hulman-institute-of-technology in Physics 2 at Rose-Hulman Institute of Technology.


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Date Created: 10/19/15
RKZ integration MM June 27 2006 rev cl T 39 39 39 39 enital motion ofapanicle launched from a height y with an initial velocity VW subject to gravity and acceleration due to air Controls lor Lime stop and air drag The time step in the numerical integration is adjustable via ascroll bar or 39slider39 This slider has a 39linked cell39 G1 and the time step dt in cell B6 is the product ofcells G1 and G2 A a second slider controls the acceleration due to air drag with a linked cell G3 The acceleration coetficient due to drag in cell B7 is the product ofcells G3 and G4 Naming cells To help interpret the calculations a cells B3B7 have been 39named39 Cells A3A7 have text in them asalabeliu 39 quot UL F L L B4 has been selected and it has been given the name Vyo as shown in theAddress Wmduw at the upper left this box would say B4 ifthis cell had not been named Naming a cell is most easily done by rst 39 39 L quot clicking one 39 39 39 You can also select the cell and then on the top menu go to Insert Name De ne to name this cell Ifyou mess up d ame cell A3 as yo Instead ofnaming B3 as yo Naturally I have never made this silly mistakel then nii39 quot 39 going to A L 39 hiiii ii r cell A37 then hit etrlc to copy it Then select cell B3 click on theAddrsss Wmduw and hit Ctrl This pastes ya in the upper left box as the name ofcell B3 and you only typed it once 381 time step hnked cali 1 00305 utrnultipliar A a RKZ vertical motion time step Shaw1 yo 1 diaaslidei 1 Vyo 4 9 g 8 1 drag 1mk2d call 1 OOELOS drag multiplier a drag 1 00E 03 ULL STEP HALF STEP time W a v 9 79 82401 1 009335 4 881285 79 82383 0 00381 1 0718598 4 862571 79 82364 1 027861 4 843857 79 82346 0 00762 1 037053 4 825144 79 82328 1 046245 4 80643 79 8231 1 I RungeKutm 2nd order RKZ calculations L 39LL quotr quot andvelocit e average acceleration is anvgAvAt Using the average acceleration we could calculate the velocity at atime At later as Since vtAt vt aivg At 1 And likewise vieg AyAt so that ytAt vt vwg At 2 This is pretty much what we will do with one small wrinkle First we will 39halfstep39 the position and velocity to the middle ofthe time inteival vtAt2 vt at AtZ and ytAt2 yt vt AtZ With the halfstep values ofposition and velocity we can calculate the acceleration M in the middle of the time step Then we use Eqs 1 and 2 to do afull time step ofposition and velocity t i dually L or idragmultiplied byvabsv I39 39 i u uuc l lDEFlGlHll 3 llimesteplmkedcell l 00E 5 dt multiplier 1 may linked cell 1 00E 3 drag multiplier A B e RKZ verti al m lion limeste siihsiAJ WW NJWW WA g aidragC11absC11 B11dt2C11 C11dt2D11 g aidragF11absF11 A a 7 RKZ vemcal motlon 7 m3 slepflldejl 7 38 me step lmked 92117 if 1 00E705 dtmultlpller 77w 1 mansllgey t r 1 my 1mm 2 JVyo 4 9 1 OOEVOS drag multlpller g r 8 dt 0 00 afdrag 1 00303 if ULL STEP HALF STEP We y w y w 7 0 9 79 82401 1 009335 4 881285 79 82383 70700381 1 018598 74 802571 79 82304 1 027801 74 843857 797782346 2 000702 1037053 4825144 79 82328 1040245 480043 79 8231 shortcuts to save typing For 39 A10dt39 type the 3939 then click on cell A10 type the w and click on the cell named olt ll in B10 C10 and type in D10 t E10 an use the dmg button to 39drag39 it to F10 copy D10 to 310 type B11 and use the dmg button to 39oltag39 it to C11 select D10G10 and 39drag39 down to D11G11 Finally Select A11G11 and 39drag39 down as far as you want Maybe 200300 rows Then plot y vs t or Vy vs t Im portant You must set your security level to medium or something so the sliders will run Go to Tools on the top menu then Options then the Security tab then Macro Security toward the bottom of that popup then select the security level


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