Given data:
Amplitude (A) = 2 cm
Frequency (f) = 100 Hz
Speed of wave (v) = 30 cm/s
Time (t) = 3 seconds
Distance from origin (x) = ?

Calculating wavelength:
The speed of a wave is given by the equation v = λf, where λ is the wavelength.
So, we can rearrange the equation to solve for the wavelength:
λ = v/f = 30 cm/s / 100 Hz = 0.3 cm

Calculating displacement:
The general equation for a simple harmonic wave is given by the equation y = A sin(kx - ωt), where y is the displacement, A is the amplitude, k is the wave number, x is the position of the particle, ω is the angular frequency, and t is the time.

Since the wave is traveling at a speed of 30 cm/s, we can write the equation as y = A sin(kx - ωt) = A sin(k(x - vt)), where v is the speed of the wave.

At t = 3 seconds, the displacement can be calculated as:
y = A sin(k(x - vt)) = 2 cm sin(k(x - 30 cm/s * 3 s))
To find the displacement, we need to find the value of k(x - 90 cm).

Calculating wave number:
The wave number is given by the equation k = 2π/λ, where λ is the wavelength.
So, we can calculate the wave number using the given wavelength:
k = 2π/0.3 cm ≈ 20.94 cm^(-1)

Calculating displacement (continued):
Now we can find the displacement at t = 3 seconds:
y = 2 cm sin(20.94 cm^(-1) * (x - 90 cm))
Since the particle is away from the origin, the displacement will be non-zero. Let's assume the displacement is y cm.

Solving for displacement:
To find the value of y, we need to solve the equation:
2 cm sin(20.94 cm^(-1) * (x - 90 cm)) = y

Calculating velocity:
The velocity of a particle in a simple harmonic wave is given by the equation v = ωA cos(kx - ωt), where v is the velocity, A is the amplitude, k is the wave number, x is the position of the particle, ω is the angular frequency, and t is the time.

At t = 3 seconds, the velocity can be calculated as:
v = ωA cos(kx - ωt) = ωA cos(k(x - vt))
To find the velocity, we need to find the value of k(x - 90 cm).

Calculating angular frequency:
The angular frequency is given by the equation ω = 2πf, where f is the frequency.
So, we can calculate the angular frequency using the given frequency:
ω = 2π * 100 Hz = 200π rad/s

Calculating velocity (continued):
Now we can find the velocity at t = 3 seconds:
v = 200π rad/s *