As with any question that lists numbers, the first thing to do is to note down the known variables:
f = 320 Hz
A = 0.50 mm
x = 0 mm
v = ?
This requires use of two of the simple harmonic motion (SHM) equations: x = Asinwt and v = Awcoswt. The second equation is simply the derivative of the first equation. Firstly, substitute the known values into the first equation to find t:
x = Asinwt
0 = 0.50 * sinwt
w = 2pif (definition of angular frequency)
0 = 0.50 * sin(2pi320)t
This gives (2pi320)t = k*pi, where k is a whole number. This is because sine of any multiple of pi will give zero, which can be seen from the graph of sine.
This gives t = k / 2*320 = k / 640.
Substitute this value of t into the equation for v to find the speed:
v = Awcoswt
= 0.50 * (2pi320) * cos((2pi320)*(k/640))
= pi * 320 * cos(k*pi)
The cosine of any multiple of pi is either +1 or -1. Since we only need the speed of the tip, whether it is + or - doesn't matter. Therefore:
v = pi * 320 * 1
= 320pi mm/s