Mini-Application\tProject Falling\tBodies Overview Reference:\tWatch\tthe\tvideo\thttps://www.youtube.com/watch?v=TMQdGqQuY5w Sir\tIsaac\tNewton\tstated\tthree\tbasic\tlaws\tof\tmotion.\tThe\tsecond\tlaw\tstate\tthat\tforce\t(F)\tis\tequal to\tmass\t(m)\ttimes\tthe\tacceleration\t(a),\tthat\tis,\tF=ma.\tSince\tthe\tacceleration\tis\tthe\tsecond derivative\tof\tthe\tposition,\twe\tcan\tuse\tthis\tto\twrite\tsecond\torder\tdifferential\tequations\ton\tthe motion\tof\tan\tobject.\tIn\tsome\tproblems,\twe\tcan\tnote\tthat\tacceleration\tis\tthe\tfirst\tderivative\tof the\tvelocity\tand\twrite\tan\tequation\tof\tmotion\tfor\tthe\tvelocity\tthat\tis\ta\tfirst\torder\tequation. Problems Consider\tthe\tproblem\tof\tdropping\tan\tobject\tfrom\ta\thigh\tbridge.\tWe'll\tconsider\ttwo\tproblems: (1)\tno\tair\tresistance\ton\tthe\tfalling\tbody,\tand\t(2)\tthe\teffect\tof\tair\tresistance\tdrag\ton\tthe\tobject. Figure\t1\t-\tFalling\tbody\t-\tdropping\tan\tobject\tfrom\ta\tbridge. 1. (4\tpts)\tWrite\tand\tsolve\ta\tdifferential\tequation\tfor\tthe\tfalling\tbody\twithout\tair\tresistance (that\tis,\tno\tdrag).\tNote\tthat\tthe\tonly\tforce\tacting\ton\tthe\tbody\tis\tits\tweight\tdue\tto\tgravity, that\tis,\tW=mg\twhere\tg\tis\tthe\tacceleration\tdue\tto\t(Earth's)\tgravity\tand\tis\tequal\tto\t-32 feet/second2\tor\t-9.8\tmeters\tper\tsecond\tsquared.\tDerive\tequations\tfor\tboth\tthe\tposition\tand velocity.\tNote\tthat\tthe\tmotion\tbecomes\tindependent\tof\tthe\tmass\t(as\tdemonstrated\tby Galileo\tat\tthe\tLeaning\tTower\tof\tPisa).\tFor\tspecifics,\tuse\tthat\tthe\theight\tof\tthe\tbridge\tis\t500 feet.\t(For\tthe\tcoordinate\tsystem,\ttake\tthe\theight\ty\tas\tpositive\tin\tthe\tdownward\tdirection with\tthe\tbridge\tat\ty=0.) 2. (6\tpts)\tNote\tthat\tthe\tgravitational\tacceleration\tconstant\tg\tacts\tas\ta\t\"parameter\"\tfor\tthis problem.\tThat\tis,\tit\tis\tconstant\tfor\tthe\tcontext\tof\ta\tproblem\ton\tEarth,\tbut\twould\tchange from\tplanet\tto\tplanet\t(that\tis,\tyour\tweight\tis\tdifferent\ton\tdifferent\tplanets.)\tSolve\tthe\tsame equations\tfor\tboth\tthe\tmoon\tand\tMars.\tCan\tyou\tdevelop\ta\t\"closed\tform\"\tsolution\tin\tterms of\tthis\tparameter?\t(That\tis,\ta\tsolution\tin\twhich\tyou\tcan\tplug\tin\tyour\tvalue\tof\tthe gravitational\tacceleration\tand\tget\tthe\tcorrect\tequations\tfor\tthe\tposition\tand\tvelocity without\tresolving\tthe\tdifferential\tequation.) 3. (7\tpts)\tNow\tsolve\tthe\tsame\tproblem\tbut\twith\tair\tresistance.\tThe\tair\tresistance\tis\tmodeled\tas a\tdrag\tterm\twhose\tforce\tis\tproportional\tto\tthe\tvelocity\t(see\tthe\tvideo\tor\tsimilar\treferences.) Use\tthat\tthe\tacceleration\tis\tthe\tfirst\tderivative\tof\tthe\tvelocity\tto\twrite\tand\tsolve\ta\tfirst\torder equation\t(again,\tsee\tthe\treference).\tAs\tin\tthe\tvideo,\tuse\tthat\tthe\tdrag\tconstant\tis\tk=0.125. 4. (8\tpts)\tNote\tthat\tthe\tequations\tnow\thave\tthree\tparameters\t-\tthe\tmass\tm,\tthe\tgravitational acceleration\tg,\tand\tthe\tdrag\tcoefficient\tk.\tExamine\tthe\teffects\tof\tthese\tparameters\ton\tthe solution.\tSee\tif\tyou\tcan\tfind\ta\tclosed\tform\tsolution\tin\tterms\tof\tthese\tparameters.\tGraph\tthe position\tand\tvelocity\tfor\texample\tvalues\tof\tthe\tparameters.\tWhat\tdo\tyou\tnotice\tabout\tthe velocity?\tAs\ta\ttest\tcase,\tfind\tsuitable\tvalues\tof\tthese\tparameters\tfor\tMars\tand\tcompare\tthe solution\twith\tthat\tof\tEarth