After all, it does change direction when reflection occurs. So shouldn't it also accelerate? And since the acceleration cannot increase the speed of light, mustn't it slow down?
Answer
Light does not slow down during a reflection.
Light is a signal disturbance in electric and magnetic fields. These disturbances propagate through space at a fixed speed $c$ in vacuum. The situation is completely analogous, in a mathematical sense, to a wave pulse that is sent along a string. When the pulse encounters a boundary, it flips direction, and may or may not change phase depending on the type of boundary encountered. For good graphical depictions of this phenomenon, visit this page.
If you emit a pulse of light at a distance of 1 meter from a plane mirror, and measure the amount of time it takes for the signal to return, you will find that it is 2 meters / $c$, neglecting refractive effects of the air. In this sense, we say that the light has not slowed down, even though it has changed direction in the middle of its journey.
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