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Title: Trigonometric Functions
Trigonometric Functions
A swimmer on a floating air mattress will follow the rise and fall of the waves in lake. When the person is on the top of the wave her feet are 0.5m above the average surface height of the lake. When she is in a trough her feet dig 0.5m into the water (-0.5m). The peaks of the waves are separated by two seconds. Create an equation of the sinusoidal function that models this movement assuming that she starts at the average surface height of the lake and heads upwards.
Answer:
Start off with the general form for the displacement (s) of a sinusoidal function:
s = A.sin(ωt + ϕ)
where A is the amplitude of the wave; t =time; ω is the angular frequency and ϕ represents the phase angle.
Using the given information:
"When the person is on the top of the wave her feet are 0.5m above the average surface height of the lake."
So (A) = 0.5 m
s = 0.5.sin(ωt + ϕ)
" ... assuming that she starts at the average surface height of the lake and heads upwards."
So when (t = 0), (s) = 0
0 = 0.5.sin(0 + ϕ)
So sin(ϕ)
So (ϕ) = 0
s = 0.5.sin(ωt)
So now we need to deal with (ω)
ω = 2πf
Where (f) is the frequency of the wave
f = 1 / T
where T is the periodic time of the wave
So:
ω = 2π/T
"The peaks of the waves are separated by two seconds."
So T = 2 seconds
ω = (2π / 2) = π radians/s
s = 0.5.sin(πt) metres
It's arguably better to leave it as (π) rather than putting in a numeric value. Your choice.
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