MA 36600 LECTURE NOTES: FRIDAY, JANUARY 16 Solutions of Some Differential Equations Gravity without Air Resistance. Say that we have a mass m at a position x = x ( t ). We wish to discuss its motion under the infuence oF gravity. We assume For simplicity that the object is near sea level, at the surFace oF the Earth. #1: Our variables are time t and position x . Note that t is an independent variable, and x = x ( t )isa dependent variable. #2: The principle that underlies the problem is Newton’s Law oF Gravity i.e. F = − mg . (This is only true For a mass that is near sea level!) #3: Our initial conditions are x (0) = x0 and v (0) = v0 . The initial value problem is m d 2 x dt 2 = − mg ° Di±erential Equation x (0) = x0 v (0) = v0 ° Initial Conditions We will show that the solution is x ( t )= x0 + v0 t − 1 2 gt 2 . To this end, ²rst we consider the velocity v = x ° ( t ). We have the initial value problem dv dt = − g ° Di±erential Equation v (0) = v0 ° Initial Condition The di±erential equation is dv dt = − g = ⇒ v ( t )= − gt + C For some constant C . (You can see this by considering
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.