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Unformatted text preview: Physics 318: Problem Set 1 Due Wednesday, Jan 30, 2008 1. A particle is projected vertically upward in a constant gravitational field with an initial velocity v . Show that if there is a retarding force proportional to the square of the instantaneous velocity, then the velocity of the particle when it returns to the initial position is v v t radicalbig v 2 + v 2 t , where v t is the terminal velocity which would be attained by a freely falling particle. [Hint: change the dependent variable in the differential equation from time t to distance z using dv/dt = ( dv/dz )( dz/dt ) = vdv/dz .] 2. Spherical polar coordinates ( r,, ) are defined by x = r sin cos , y = r sin sin , z = r cos , where r 0, 0 and 0 2 . a. Using the chain rule, show that the vector d r = dx e x + dy e y + dz e z can be written as dr e r + rd e + r sin d e , where e r , e and e are unit vectors. Find the explicit form of these vectors on the basis e x ,...
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