# prob3 - Problems 3-2 1 A body of weight 32 lb is dropped...

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Problems 3-2 1. A body of weight 32 lb is dropped from a height of 100 ft in a medium offering an air resistance proportional to the velocity. If the limiting velocity is 400 ft/sec, find the velocity and displacement at any time. Find the time at which the velocity is 200 ft/sec. 2. A freely falling body of weight lb encounters an air resistance equal in pounds to where is the speed. Set up the differential equation for the velocity. Without solving the equation, find the limiting velocity. 3. Solve: DE: IC: and show that the limiting velocity is . 4. Repeat Prob. 1 with air resistance proportional to the square of the velocity. 5. A body of mass m is thrown vertically into the air with a speed . The body encounters an air resistance proportional to the velocity. Set up and solve the differential equation for the velocity of the body while it is rising. At what time does the body reach its maximum height? 6. A particle of mass moves along the axis subject to a resisting force proportional to the cube of the velocity. Find the velocity and the displacement in terms of the time t, the initial velocity, and the initial displacement. Answers and hints: 1. when 2. ft/sec 3. 4. when 5. DE: IC: when max height 6.

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Problems 3-3 1. A particle of unit mass moves along the axis and is acted on by a force . If the particle starts from the rest at , find its position at any time. Also find the time for the particle to reach the origin. 2. A particle of unit mass moves along the axis and is acted on by a force . If the particle starts at with a velocity , find the position of the particle at any time and the limiting position as time increases. 3. If a pendulum hanging at rest is given an initial velocity , how high will it rise? (See Sec. 1-3, Prob. 3.) 4. A particle starts from rest at and moves along the axis under the force . Show that the time to reach the origin is Evaluate this integral in terms of the gamma function (see Prob. 6, page 31) to obtain . Answers and Hints: 1. when 2. as 3. Problems 3-4 1. One end of a 3-ft chain is held so that the other end just touches the floor. It is then released. Find the force with which the high end hits the ground. 2. A chain is coiled up on the ground. One end is lifted with a constant force. Find the velocity. 3. A raindrop falls through dry air. The raindrop is assumed to lose mass at a constant rate by evaporation. Assume that the evaporating mass has the same velocity as the raindrop. If the air resistance is proportional to the velocity, find the velocity of the raindrop.
4. A spherical raindrop falls from the rest through dry air. The raindrop loses mass at a rate proportional to its surface area. Find the velocity of the raindrop at any time, ignoring air

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