Question

An iron ball is bobbing up and down on the end of a spring. The maximum height of the ball is 12 feet and its minimum height is 2 feet. It takes the ball 2 seconds to go from its maximum height to its minimum height.

Which model best represents the height, h, of the ball after t seconds?

Answer 1

The model that best represents the height, h, of the ball after t seconds is a sinusoidal function. The general form of a sinusoidal function is:

h(t) = A + B*sin(C(t - D))

Where:
- A is the vertical shift of the function (in this case, it would be the average height of the ball which is the midpoint between the maximum and minimum heights, so A = (12+2)/2 = 7 ft)
- B is the amplitude of the function (half the difference between the maximum and minimum heights, so B = (12-2)/2 = 5 ft)
- C is the frequency of the function, which is related to the period of the function (in this case, the ball takes 2 seconds to go from maximum to minimum height, so the period would be 2 seconds. Therefore, C = 2π/(period) = 2π/2 = π)
- D is the horizontal shift of the function (it represents the phase shift, in this case, there's no phase shift, so D = 0)

Therefore, the equation that best represents the height, h, of the ball after t seconds is:

h(t) = 7 + 5*sin(πt)