A buoy floating in the ocean is bobbing in simple harmonic motion with amplitude 7ft and period 8 seconds. Its displacement d from sea level at time t=0 seconds is 0 ft, and initially it moves upward. (Note that upward is the positive direction.)
Give the equation modeling the displacement d as a function of time t.
To model the displacement d of the buoy as a function of time, we can use the formula for simple harmonic motion:
d(t) = A * sin((2π/T) * t + φ)
Where:
- d(t) represents the displacement of the buoy at time t.
- A is the amplitude of the motion, which is given as 7 ft in this case.
- T is the period of the motion, given as 8 seconds.
- t is the time variable.
- φ is the phase angle, which determines the starting position of the motion.
Since the buoy starts from the equilibrium position (d = 0) and moves upward initially, the phase angle φ will be 0.
Thus, by substituting the given values into the equation, we have:
d(t) = 7 * sin((2π/8) * t)
Simplifying further:
d(t) = 7 * sin((π/4) * t)
Therefore, the equation that models the displacement of the buoy as a function of time is d(t) = 7 * sin((π/4) * t).
To find the equation that models the displacement d as a function of time t, we can use the formula for simple harmonic motion:
d(t) = A * cos(2π * t / T) + c
Where:
- d(t) represents the displacement at time t.
- A is the amplitude of the motion.
- T is the period of the motion.
- c is the vertical shift (initial displacement from sea level).
In this case:
- A = 7 ft (amplitude)
- T = 8 s (period)
- c = 0 ft (initial displacement)
Substituting the values into the formula, we have:
d(t) = 7 * cos(2π * t / 8) + 0
Simplifying further:
d(t) = 7 * cos(π * t / 4)
Thus, the equation that models the displacement d as a function of time t is:
d(t) = 7 * cos(π * t / 4)