A hollow steel ball weighing 4 pounds is suspended from a spring. This stretches the spring 1515 feet. The ball is started in motion from the equilibrium position with a downward velocity of 88 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 the positive direction is down.) Take as the gravitational acceleration 32 feet per second per second.

Explanation:

Given that:

The weight of ball = 4 pounds

The spring stretch  x = 1/15 feet

Using the relation of weight on an object:

W = mg

m = W/g

m = 4 / 32

m = 1/8

Now, from Hooke's law:

F = kx

4 =k(1/5)

k = 20 lb/ft

However, since the air resistance is 4 times the velocity;

Then, we can say:

C = 4

Now, for the damped vibration in the spring-mass system, we have:

Solving the differential equation:

m² + 32m + 160 = 0

Solving the equation:

m = -25.80 or m = -6.20

So, the general solution for the equation is:

y(0) = 0   ;  y'(0) = 8

At y'(0) = 8

From (1), let , then replace the value of c_1 into equation (2)

From

∴

The required solution in terms of t is:

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