Answer Happy

OBJECTIVE Suppose that you are standing on a bridge above the Malad River campon Suppoes that you plan to jump off that beide You hide wish. Instead, you plan to attack a bonge cond to your feet, to dive gally into the vid and to be pulled back gotly by the cord before you hit the river that is 53 meters below. You have height awal different chords with which to as your feet, including several standard bunge coeds, a climbing pe, and a stal cable You need to choom the stoms and length of the cond as to avoid the upciated with an unexpected water landing. You are undaunted by this Each of the cords you have brought will be tied off so as to be 30 meters long when hanging from the bridge Call the position at the bottom of the cand 0, and measure the position of your feet below that atural longh), whes increase as you go down and is a function of une t. Then at the time you Jump (0) -30, while if your 2 meter frame hits the water head first, them at that time ()-53-30-2 You know that the acceleration due to gravity is a constant, called g, so that the force pulling downwards your body is reg. You know that when you leap from the bridge, air resistance will increase proportionally to your spood, providing a force in the opposite direction to your motion of about av, where a la a consta andela your velocity. Finally, you know that Hooke's law describing the action of springs says that the bune cond will eventually court a force on you proportional to its distance past its natural length. Thus, you know that hece of the cond pulling you back from destruction may be expresand K(x) => 450 ₁ => The number is called the spring matant, and is where the stiffs of the coed you use influences the gation. For example, if you used the stoel abie, then & would be very large, giving a tremendous stopping forme very saddenly as you passed the natural length of the cable. This could lead to discomfort, injury, or even a Darwis award. You want to choose the cord with a & value large enough to stop you above or just touching the water, but not too suddenly Consequently, you are interested in find the distance you fall below the natural length of the cord as a function of the spring constant. To do that, you must solve the differential equation that we have derived in words above the force on your body is described by the mumy+b(a)-ax². Here is your man, and a' is the rate of change of your position below the equilibrium with respect to time, Le your velocity. The constanta is for air rmistance depends on a number of things, including whether you wear your skin-tight plak spandex or your parachute pants and XXL Ocean Surfari T-shirt, but you know the value soday is 2.8 This is a nonlinear differential equation, because of the M(x) term. However, inside this nonlinear equation are two linear equations that differ when <0 and 20. We will solve these two separately, and then piece the solution together at a(t)-0. On the following page, are a list of 7 exercises for you to complete on your own. #2 and #5 require you to solve a second oeder linear differential equation with constant coefficients. For #2 you may use either Variation of Parameters, or the Laplace Transform to find your solution. For #5, you will need to use the Laplace Trandonn. For #3, #4, #6, and #7 you will use your own ma (in kg). 5. Using the Laplace transform, we want to solve the second part of the initial value problem when the bungee jumper is 30 or more foet below the bridge. That is, we want to solve the following IVP using the Laplace Transform. me+as-by)-mg; for 12 f (₁)-0₁ √(41)=52₂. Since the Laplace transform requires to know the value of a() at t-0, we will define a new variable t-t, and a new function ()-(+₁) Notice that this is just applying a horisontal shift to y, which will not change it's derivatives. Thus yo would satisfy the same differential equation, but have the following initial conditions, my+a+mg. 35(0)-(₁)-0; (0)-(4₁)-5. We will solve this shifted initial value problem for y() using the Laplace transform, then apply ()=(+₁)=za(t). Again, you may use o-2.8 and g-9.8, but leave m and kas unknown constants. The solution ay(t) represents your position below the natural length of the cond after is starts to pull back. (I recommend that you lease vy, a, and g as variables when find the solution to the IVP, and only rubstitute the values of these three variables at the end.)
OBJECTIVE Support that you with all those off the bride Yowe will plan to the date the world by the below. You have maddede with which to your Badead rom, and welche Youthwest to avoid the pled with add wa Yound by Task you! Each of the contes you have brought will be tiedot to be with bobo Che point the bottom of the cord de pour b) where you go down the Jump (0) = 30 for the start in that)-5-30-2 You know that there to gravity is a contatt, culled that the force on your body w my. You know that you hap from the bridge will be to your providing for the direction to your of about and you wilay Pally, you know that He low crng the time the bon cord will tally eat for you proportional to tadatence ple The you know that force of the card paling you back from date may be M) 0.30 -> The sumber is called the content, and is where the time of the cand youth equation example, if you wand the table, the would be very loving to stop force veryoddenly you the stumth of the cable This could only or even Darwin award. You want to choose the cond with a large enough to stop you show or Just touching the but not too wuddenly. Countyoused in the day fall below the naturalmth of the cards a function of the spring co. To do that you the differential equation that we have derived in woede above the force on your body is described by the quation "M) - Here is your and the rate change of your position below the limithet to me Le your velocity. The content is for sirritance depende sa umber of thing, inding the you wear your skintight pink spandex or your parachute pont and XXL One Surlari bol you know the value today is 28 This is a nonlinear differential equation, because of the a) term. However, ide the most are two linea equations that differ wh: < and > 0. We will solve the two way, and the place the solution together at t) -0. On the following page are a list of 7 for you to complete on your own. 2 dequire you to solve a second order linear differential equation with constant code or you may with Variation of Parameters, or the Laplace Transform to find your solution for you will need to the Laplace Transform. For 4, and you will come your own man 3. Using the Laplace trataform, www to solve the second part of the initial de problem when the bungee jumper is 30 or more foet below the bridge. That is, we want to solve the following IVPwg the Laplace Transform. +- M-) - mfort Since the Laplace transform requires to now the value of a(t) we will defines variable =t- and a new function () (+) Notice that this is just applying a horontals to which will not change it's derivative Thas a would stify the edital equation, but have the following initial conditions my+o+lym (0) =) = 0; (0) (0) +. We will solve this shifted initial value problem foru) uning the Laplace transform, then apply () ==3(+1) ==>(4). Apun, you may use a 28 and 9 = 38, but nevem and i w ukos constants. The solution (t) represents your position below the satural length of the orders starts to pull back. (Trecommend that you love , a, andy a variables when find the solution to the IVP, and only latitude the soluts of these three variables at the end)