# Amplitude speed of a vibrating string

Interpreted in degree mode, the equation y = cos(72000t) sin(2x) models the motion of a stretched string that is 180 centimeters long (x = 0 is one end of the string and x = 180 is the other), and that is vibrating 200 times per second. The times when the model predicts that the string is straight whenever T = x/800 is an odd x. T is 1/800, 3/800 are answers.

The center of the string never moves and the point on the string that is moving the fastest at an instant when the string appears straight are at x = 45 and 135.

Modeling in excel provides the above.

How fast is this point moving? I know the answer is 1257 cm/sec but I don't know what the equation is to get that answer. Been looking around the internet without much luck. Wave motion deals with the speed of the wave along the x-axis but I'm trying to determine speed along the y-axis when the string is straight at point 45. Any tips much appreciated!

You just have to calculate $$\frac{\partial y}{\partial t}$$ this gives you the speed along the y-axis.
With $\omega=72000$ and $k=2$: $$\frac{\partial y}{\partial t}=\frac{\partial (\cos (\omega t)\sin(kx))}{\partial t}=-\omega\sin (\omega t)\sin(kx)$$
So the answer should be $\omega$. But we cannot use the value $72000$, it is in sexagesimal degrees. We know that is vibrating 200 times per second, so $$f=\frac{\omega}{2\pi}$$ $$200=\frac{\omega}{2\pi}$$ $$\omega=400\pi\approx 1257$$