Problem 1A-8**: Falling Ball with Air Resistance

A ball is dropped from a height H. The function y(t) denotes the vertical position of the ball at time t, and at the moment the ball is released t is set to 0. The motion of the ball can be described by the differential equation

where G=g/2 (g is a given constant whose meaning will become clear later) and ε are positive constants. The first term on the right-hand side of the equation is due to the gravitational pull, and the second term is due to the air resistance. Determine y(t) and its speed, v(t), completely in terms of given constants H, G and ε. What is the terminal speed in this case?

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The equation of motion is

Since the speed is defined as v(t)=dx/dt, the equation of motion can be written as

Integrating both side of Eq.(2), and utilizing the relations

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