MA 36600 LECTURE NOTES: FRIDAY, JANUARY 16Solutions of Some Differential EquationsGravity without Air Resistance.Say that we have a massmat a positionx=x(t). We wish to discussits motion under the influence of gravity. We assume for simplicity that the object is near sea level, at thesurface of the Earth.#1: Our variables are timetand positionx. Note thattis an independent variable, andx=x(t) is adependent variable.#2: The principle that underlies the problem is Newton’s Law of Gravity i.e.F=−m g. (This is onlytrue for a mass that is near sea level!)#3: Our initial conditions arex(0) =x0andv(0) =v0.The initial value problem ismd2xdt2=−m gDifferentialEquationx(0) =x0v(0) =v0InitialConditionsWe will show that the solution isx(t) =x0+v0t−12g t2.To this end, first we consider the velocityv=x(t). We have the initial value problemdvdt=−gDifferentialEquationv(0) =v0InitialConditionThe differential equation isdvdt=−g=⇒v(t) =−g t+Cfor some constantC. (You can see this by considering
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