Question

Derive an equation d = asin(bt) for the displacement d (in feet) of a buoy in relation to sea level over t seconds. (assume the buoy starts with a displacement of 0 and then goes up to its maximum height.) the maximum displacement of the buoy is 1.5 feet in either direction, and the time it takes for the buoy to go from its highest point to its lowest point is 3 seconds.

use the period to determine b.
b =

Answer 1

To derive the equation d = asin(bt) for the displacement of a buoy in relation to sea level over t seconds, we can use the given information about the maximum displacement and period. By using the equation for simple harmonic motion, we can determine the value of b.

Step-by-step explanation:

To derive the equation d = asin(bt) for the displacement of a buoy in relation to sea level over t seconds, we can start by understanding the given information. The maximum displacement of the buoy is 1.5 feet in either direction, and the time it takes for the buoy to go from its highest point to its lowest point is 3 seconds. We are asked to derive the equation using the period T.

The period T is the time it takes for one complete cycle of the buoy's motion. In this case, the buoy goes from its highest point to its lowest point and back to its highest point, so T is equal to 3 seconds. The period T is related to the value of b in the equation.

The equation for the displacement of a simple harmonic motion with amplitude A and period T is given by d = Asin(2πt/T). Since the maximum displacement of the buoy is 1.5 feet, we can set A = 1.5. Plugging in these values and solving for b, we get b = 2π/T.