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Unformatted text preview: Problem Set 1: Problem 6.
Problem: Earth's gravitational acceleration on an object is a = gR2 /r2 , where r is distance from Earth's center, R is Earth's radius and g is standard gravitational acceleration at the planet's surface. (a) Compute the velocity, v, of an object launched from the North Pole with initial velocity vo as a function of vo , g, r and R. (b) For what value of vo will the object escape Earth's gravitational pull, i.e., for what value of vo does v 0 as r ? (c) Determine the value of vo computed in Part (b). Earth's radius is R = 3960 mi. Express your answer in ft/sec.
. .. . ......... .......... .. . . . R . .. ........... .......... . . . . . . ........................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... ....................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... . .. ... .................. . ...................... vo a . . v . .. . ................................................... ..................................................... . .. . . .. .. r ........................................................................................................ . . Solution: Because the problem does not involve time, the easiest approach is to begin with the differential equation relating acceleration and velocity, viz., a=v (a) For the given acceleration, we have  Then, integrating, we have
v vo r dv dr = a dr = v dv gR2 dr = v dv r2 = v dv = gR2
v=v dr r2
r=r r=R v dv = gR2 R dr r2 = 1 2 v 2 =
v=vo gR2 r or, 1 2 2 v  vo = gR2 2 1 1  r R R r Thus, the particle's velocity as a function of its distance from Earth's center is given by
2 v 2 = vo  2gR 1  (b) The object will rise indefinitely if its velocity approaches zero only as r . From the solution of Part (a), we have 2 lim v 2 = vo  2gR
r ...
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This note was uploaded on 05/12/2010 for the course AME 301 taught by Professor Shiflett during the Spring '06 term at USC.
 Spring '06
 Shiflett

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