Your fishing bobber oscillates up and down from the current in the river in a harmonic motion. The bobber moves a total of 2.5 inches from its high point to its low point then returns up to the high point every 3 seconds. Write an equation modeling the motion of the bobber at its high point at time t = 0.
a max at t=0 indicates a cosine wave
you have the period and displacement, and there is no phase shift
just follow the standard format
To write an equation modeling the motion of the fishing bobber at its high point at time t = 0, we can use the equation for simple harmonic motion:
y(t) = A * cos(ωt + φ)
where:
y(t) represents the displacement of the bobber from its equilibrium position at time t,
A represents the amplitude of the bobber's motion,
ω represents the angular frequency, and
φ represents the phase constant.
Given that the bobber moves a total of 2.5 inches from its high point to its low point, we can conclude that the amplitude (A) is 1.25 inches (half of the total range of motion).
Now, we need to find the angular frequency (ω). The angular frequency can be calculated using the formula:
ω = 2π / T
where T represents the period of the motion. In this case, the bobber returns to its high point every 3 seconds, so T = 3 seconds. Substituting this value into the equation, we get:
ω = 2π / 3
Lastly, to determine the phase constant (φ), we are given that the bobber is at its high point at t = 0. Therefore, the initial phase is zero (φ = 0).
Putting it all together, the equation for the motion of the bobber at its high point can be written as:
y(t) = 1.25 * cos((2π / 3) * t)
This equation represents the displacement of the fishing bobber from its equilibrium position (high point) at any given time t.