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avenior

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## Homework Statement

On the axis of an infinite wedge that moves with velocity ##\vec{V}##, the body decays with the formation of a lot of splinters that fly away uniformly in all directions with velocity ##\vec{u}##. What should be the angle of the wedge that half of the splinters fall on its side surface?

## Homework Equations

Lorentz transformations

## The Attempt at a Solution

The right answer is

##\operatorname{tg}\frac{\alpha}{2} = \frac{u}{V}\sqrt{1-\frac{V^2}{c^2}}##

As I understand, the figure to this problem looks like this

If ##\varphi## is the angle between ##\vec{u}## and ##Ox##, then ##\vec{u}_{spl} = (u \cos \varphi, u \sin \varphi )##. By making the Lorentz transformations, we obtain that $$\vec{u^{'}}_{spl} = \left(\frac{u \cos \varphi - V}{1 - \frac{Vu \cos \varphi}{c^2}}, \frac{1}{\gamma} \frac{u \sin \varphi}{1 - \frac{Vu \cos \varphi}{c^2}} \right).$$

How can we take into account that the half should fall to the surface?

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