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Mechanical displacement - Steel ball problem

A steel ball weighing 128 pounds (mass= 4 slugs) is suspended from a spring. This stretches the spring 128/485 feet.

The ball is started in motion from the equilibrium position with a downward velocity of 9 feet per second.
The air resistance (in pounds) of the moving ball numerically equals 4 times its velocity (in feet per second) .

Suppose that after t seconds the ball is y feet below its rest position. Find y in terms of t. (Note that this means that the positive direction for y is down.)

Take as the gravitational acceleration 32 feet per second per second.

When using English units (lb, ft, etc.) you need to be a bit careful with equations involving mass. Pounds (lb) is a unit of force, not mass. Using mg=F and g=32 ft/sec^2 we see that if an object weights L pounds it will have a mass of L/32 lb*sec^2/ft (or slugs).

Solution Summary

This is an example of a problem regarding mechanical displacement of a steel ball. The answer is attached in two JPG files, where the Expert has written their response using step by step calculations and diagrams to aid understanding of the problem.