In the deep ocean, a water wave with wavelength 95 m travels at 12 m/s. Suppose a small boat is at the crest of this wave, 1.8m above the equilibrium position.What will be the vertical position of the boat 3.0s later?
To determine the vertical position of the boat 3.0 seconds later, we can use the wave equation:
y = A * sin(2π/λ * (x - vt))
Here,
y = vertical position of the boat
A = amplitude (which is given as 1.8 m)
λ = wavelength (which is given as 95 m)
x = initial horizontal position of the boat (which we'll assume to be zero for simplicity)
v = speed of the wave (which is given as 12 m/s)
t = time (which is given as 3.0 s)
Now, let's substitute the given values into the equation:
y = 1.8 * sin(2π/95 * (0 - 12 * 3.0))
First, calculate the value inside the sine function:
2π/95 * (0 - 12 * 3.0) = -2π/95 * 36 = -72π/95
Now, substitute this value back into the equation:
y = 1.8 * sin(-72π/95)
To find the vertical position of the boat, we can evaluate this expression using a scientific calculator or a trigonometric table. The exact value depends on the accuracy required for the answer.
Note: It's important to keep in mind that this calculation assumes that the wave is sinusoidal and that the boat is experiencing a simple harmonic motion in sync with the wave. In reality, factors like dampening, wave shape, and the boat's motion relative to the wave may complicate the situation.