Homework 2 - m varies with the distance x as v x = αx n...

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Homework Set 2 Due: Friday, September 14, 2012 1. Consider a rocket subject to a linear resistive force, f = - bv , but no other external forces. Show that if the rocket starts from rest and ejects mass at a constant rate k = - ˙ m , then its speed is given by v = k b v ex bracketleftBigg 1 - parenleftbigg m m 0 parenrightbigg b/k bracketrightBigg , where v ex is the exhaust speed and m 0 is the initial mass of the rocket. 2. A particle of mass m moves in a medium under the influence of a retarding force f = mk ( v 3 + a 2 v ), where k and a are constants. Show that for any value of the initial speed the particle will never travel a distance greater than π/ 2 ka and that the particle only comes to rest for t → ∞ . 3. A particle of mass m = 1 kg is subject to a one-dimensional force F ( t ) = kte - αt where k = 1 N/s and α = 0 . 5 s - 1 . If the particle is initially at rest, calculate and plot with the aid of Mathematica the position, speed, and acceleration of the particle as a function of time.
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Unformatted text preview: m varies with the distance x as v ( x ) = αx n . Assume v ( x = 0) = 0 at t = 0. ( a ) Find the force, F ( x ), responsible. ( b ) Determine x ( t ) and ( c ) F ( t ). 5. A beach ball is thrown upward with initial speed v . Assume that the drag force from the air is f =-mαv . What is the speed of the ball right before it hits the ground? Does the ball spend more time or less time in the air than it would if it were thrown in vacuum? 6. Consider an object that is coasting horizontally (positive x direction) subject to a drag force f =-bv-cv 2 . Write down Newton’s second law for this object and solve for v by separating variables. Sketch the behavior of v as a function of t . Explain the time dependence for large t . (Which force term is dominant when t is large?)...
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