hw1_solns

hw1_solns - ME 17 Spring 2007 Homework #1 Solutions Problem...

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ME 17 Spring 2007 Homework #1 Solutions Problem 1 (10 pts) (a) [1pt] Verify that v (0) = 0. Start with the equations given: () tanh d d cg mg vt t cm ⎛⎞ = ⎝⎠ , where tanh( ) xx ee x = + Substitute t = 0 into the expression for v(t): 00 0 (0) tanh 0 0 2 dd d mg mg e e mg v cc e e c === + = (b) [9pts] Using v(t) as given by (2), show by explicit calculation that dv/dt is equal to equation (1). The general idea is to independently calculate the left-hand side (LHS) and right-hand side (RHS) of equation (1), and then show that they are equal. First calculate what the LHS should be if v(t) is a valid solution. Given: d d mg t = , and tanh( ) x = + Calculate the derivative of tanh(x): 22 2 2( 2 )4 1 tanh( ) cosh ( ) de e e e x dx x −− +− ++ −+ == = + 2 = So, by taking a derivative of the above v(t) , we get: 1 cosh cosh d d dm g g dt c m tt mm ⎜⎟ Is this expression equivalent to plugging the claimed solution for v(t) into the differential equation (1)? Plug the claimed solution for v(t) into the RHS of equation (1): 2 d c dv gv dt m =− (1) 2 2 tanh 1 tanh d g dv mg gt g dt m c m m = d c g t Now check if the two are equivalent: ? 2 2 1t a n h cosh d d g m t m 1 / 4 M.K.
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ME 17 Spring 2007 ? 2 2 1 1t a n h cosh d d cg t m t m ⎛⎞ =− ⎜⎟ ⎝⎠ Note that: 22 2 2 cosh sinh a n h cosh dd d d tt mm t m t m ⎛⎞ ⎛⎞ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ −= ?
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hw1_solns - ME 17 Spring 2007 Homework #1 Solutions Problem...

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