math138p_w16_asst5.pdf - PHYSICS-BASED MATH 138 Winter 2016...

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PHYSICS-BASED MATH 138 Winter 2016 Assignment 5 Topics: Parametric curves, Differential equations. Due: 2pm Friday, February 12. 1. A curve parameterized in polar coordinates is given by the vector function ( θ ) = ( r ( θ ) cos θ, r ( θ ) sin θ ) , with θ 1 θ θ 2 . (a) Show that the length of the curve is, length ( ) = θ 2 Z θ 1 s r 2 + dr 2 dθ. (b) Find the length of the polar curve r ( θ ) = 1 + cos θ , with 0 θ 2 π . 2. When a raindrop falls, it increases in size and so its mass at time t is a function of t , m ( t ) . The rate of growth of the mass is km ( t ) for some positive constant k . Applying Newton’s Law of Motion to the raindrop results in, d dt ( mv ) = gm, where v is the velocity of the raindrop (directed downward) and g is the acceleration due to gravity. The terminal velocity v ter of the raindrop is, v ter = lim t →∞ v ( t ) . Find an expression for the terminal velocity in terms of g and k . 3. Solve the following differential equations. Note you may not be able to find an explicit expression for y ( x ) in some cases.

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