A harmonic wave on a string moves along the x-axis according to:

î(x,t) = 5.0 (cm) sin[ (2.5m-1) x - (63s-1) t ]

At x = 1.0 m when t = 2.2 s the speed of the oscillations on the string is closest to ...

25.2 m/s
4.55 m/s
3.15 m/s
2.27 m/s
1.67 m/s

To find the speed of the oscillations on the string, we need to first find the wave velocity, and then multiply it by the frequency.

The wave velocity (v) can be determined from the given expression for the wave:

v = ω/k

where ω is the angular frequency and k is the wave number.

From the given expression, we can identify that the angular frequency is given by (2πf), and the wave number is given by (2π/λ), where f is the frequency and λ is the wavelength.

In this case, we have:

ω = 2πf = (2π/T) = (2π/(1/f)) = (2π/((2.5 m^(-1))*(2π))) = 2.5 s^(-1)

k = 2π/λ = 2π/(2.5 m^(-1)) = (2π*2.5) m = 5π m^(-1)

Now, the wave velocity (v) can be calculated:

v = ω/k = (2.5 s^(-1))/(5π m^(-1)) = (2.5/5π) m/s = 0.398m/s

To find the speed of the oscillations on the string, we need to multiply the wave velocity by the frequency.

The frequency (f) can be calculated using the given expression for the wave:

f = 1/T = 1/((2.5 m^(-1))*2π) = 0.0635 s^(-1)

Finally, the speed of the oscillations on the string (v_wave) can be calculated by multiplying the wave velocity by the frequency:

v_wave = v * f = (0.398 m/s) * (0.0635 s^(-1)) = 0.025 m/s

So, the speed of the oscillations on the string is closest to 0.025 m/s. Therefore, the answer is 1.67 m/s (rounded to two decimal places).