High speed and low speed photons Looking at the discovery of the neutron, and I came across this page: http://www-outreach.phy.cam.ac.uk/camphy/neutron/neutron3_1.htm 
The animation on the left, talks about low energy photons and high energy photons. And it implies the more energy a photon has the faster it moves. I don't understand the whole light is photons and waves at the same time thing, but I thought the speed of light was constant, that Gamma rays travel at the same speed as visible light, infrared, microwaves, etc.
So how is that some photons can move faster than others (with more energy)?
 A: The animation is unfortunately misleading.  The speed of light is constant and all photons, of any energy, travel at the same speed.  Higher energy photons have smaller wavelengths (or, equivalently, higher frequency) but not a different speed.  
Unfortunately, this is difficult to illustrate clearly.  The reason the illustration shows the higher energy photons as faster is because of the naturalness of equating speed with kinetic energy of an object.  It "makes sense" to most people that a more energetic particle would move faster, even if this isn't an accurate description of the phenomenon.
A: No. the energy of a photon is not linked with his speed. all photons move at the same speed $300.000 km/s$,  that corresponds to the speed of light. The different energies of pictures are associated with frequency or wavelength of this. the greater the wavelength, lower energy and vice versa. this is: $$ E = h.\nu $$
Depending on their wavelength, electromagnetic radiation is given different names, ranging from energy gamma rays (with a wavelength of about picometers) to radio waves (wavelengths of the order of kilometers) through the visible spectrum (the wavelength is in the range of tenths of micrometer).
A: In the answer by user48649 an example is given of an observation that can be interpreted as two different velocities. 

A telescope viewing a supernova from over 16 billion light years away recently clocked the low energy photon arriving 5-7 seconds later than it's high energy equivalent.

This publication shows that  actually this statement is partly true, only the distance and certainty of the conclusion is wrong. Dr Robert Wagner, of the Max-Planck-Institut für Physik, München, found that there was up to a 5 second gap between low energy and high energy photons coming from an active galactic nucleus (Markarian 501) SEVEN billion light years away. This is not to say that this is definitive proof, as Dr Wagner notes "We cannot exclude, however, the possibility that the delay we find, which is significant beyond the 95% C.L., is due to some energy-dependent effect at the source."
A: The speed of light it is medium-dependent, note $c/v= \mu$, where $\mu$ is the refractive index of the medium. But in this case, it is correct to say that a photon has a KE given by $E = hc/ \lambda$ in vacuum. At the interfaces between mediums, however, using Huygen's principle,
$\lambda$1$/ \lambda$2$= v$1$/ v$2. So light has a definite speed in a given medium, with its energy dependent on its frequency $\nu$, which is a property of the source, as $E=h\nu=hv/\lambda$.
The animation is in this case, extremely misleading.
A: All electromagnetic waves travel at the same speed in the same medium, as per
$$ v_{_{EM} }={\frac {1}{\sqrt {\mu \varepsilon }}} \tag 1 $$
It's just that more energetic waves will achieve more beats per second (higher frequency), but this means that per single beat, more energetic wave travels shorter distance, because time interval per beat decreases, as per:
$$ T= \frac 1f, \tag 2$$
So, in the end bigger beat density spatially results in a shorter wavelength, because :
$$ v_{_{EM} } = f \lambda = \text{const} \tag 3$$
must apply for (1) to be true.
A: Actually, the speed of light is not constant. A telescope viewing a supernova from over 16 billion light years away recently clocked the low energy photon arriving 5-7 seconds later than it's high energy equivalent. This was used as proof as to the fabric of space time not being smooth, but likened to a frothy foam that can affect the way photons travel. The scientist at the centre of this research was quick to note that his findings would actually undermine a long held principle of physics... that the speed of light is anything but constant!!!
