To go through one complete wiggle, the tip of the fork has to move outward 0.6mm, then 0.6mm back to the middle, then inward 0.6mm, then 0.6mm back to the middle again. So one complete wiggle of the tip moves it 2.4mm .
It does this 392 times every second. So its AVERAGE speed would be
Speed = (distance) x (frequency)
Speed = (2.4 mm) x (392 Hz)
Speed = (0.0024 m) x (392 / sec) = 0.9408 m/s .
That's the AVERAGE speed of the tip of one prong. That's easy. Sadly, the question is asking us for the MAXIMUM speed. That will be less easy.
Now right here, I'm afraid I will go off the rails for a bit ... I'm going to assert things and do things that I'm not willing to try and explain for 5 points. It may not even be correct, (which would make it a lot harder to explain). So I'm just gonna jump in and DO IT.
The way I see it, the tip of that prong is wiggling in sinusoidal wiggles. Relative to its resting position, its location is something like
x = (0.6 mm) x sin(2π x 392 t) .
and as usual, its speed is the derivative of that mess.
Speed = dx/dt = (0.6 mm) x (2π x 392) cos(2π x 392 t)
The greatest that the cosine alone can be is 1 , so the maximum value of the speed is
(0.6 mm) x ( (2π x 392))
and that's 1,477.8 mm/s or 1.4778 m/s . I think this is the answer to part-a, and now we can go on to consider the hapless fly, stuck by his pads to the wildly oscillating prong.
Part-b is easy. The fly's maximum kinetic energy is just
KE = (1/2) (flymass) (max speed)²
KE = (1/2) (0.027 g) (1.4778 m/s)²
KE = (1.35 x 10⁻⁵ kg) (2.184 m²/s²)
KE = 2.95 x 10⁻⁵ Joule
So there ya go. These are my answers and I'm stickin withum.