Why is the shadow of a wind turbine a bit slow at first then very quickly, etc. and not equivalent to the wings? When I stand at a wind turbine and look at the shadow on the ground, the shadow is a bit slow at first then very quickly, etc. Very strange.
 A: Depending on the angle of the sun, the shadow becomes elongated so that it traces out an ellipse rather than a circle. This means that the shadows of the blades must traverse a different distance across the ground but within the same time period, thus giving rise to the periodic variation in velocity.
If the sun is at an angle $\alpha$ in the sky (see the diagram below) then there are four intervals of interest:


*

*$\alpha<\frac{\pi}{4}$: the sun is low in the sky and the shadow is longer than the actual turbine. As a result, the shadow must occasionally speed up.

*$\alpha=\frac{\pi}{4}$: the dimensions of the turbine are preserved and the shadow moves at a constant speed.

*$\alpha>\frac{\pi}{4}$: the sun is high in the sky and the shadow is shorter. It must therefore occasionally slow down.

*$\alpha=\frac{\pi}{2}$: the sun is directly overhead and no shadow is cast (at least, the blades cannot be distinguished).





This is simple to tackle mathematically. When the sun is directly behind/in front of the turbine, the coordinates of a particular blade are transformed as follows when projected onto the ground:
$$ R(\cos(\theta),\sin(\theta))\mapsto R(\cos(\theta)/\tan(\alpha),\sin(\theta)) $$
as is shown above. The elongation is even greater when the sun is at a horizontal angle with respect to the normal of the blades, but I shall ignore this effect for simplicity.
While the velocity of an actual blade ($R\dot{\theta}$) is constant, its projection has a velocity that depends on the angle $\theta$,
$$ v=R\dot{\theta}\sqrt{\frac{\sin^2(\theta)}{\tan^2(\alpha)}+\cos^2(\theta)} $$
and so it follows that the shadow speed varies.
For example, here is a plot of the relative tangential velocity of the shadow as a function of $\theta$ when the sun is relatively low in the sky ($\alpha=\pi/6$):

