Lorentz factor depend on speed, but to measure the speed, we need to know the distance traveled and the times it takes to get the ratio. But according to special relativity theory, for two equivalent observer that moving relative to each other, each of them will see the other experiencing time dilation and lorentz contraction. So distance measured By one observer become smaller when seen from other observer by lorentz factor. And also time measured By one observer become longer when seen from other observer. And speed as measured By one observer must by multiplied By square of lorentz factor when calculated By other observer.
Update: With their average life time and speed, moun can reach a distance only 0.66 km, but when using time dilation formula it can reach distance 10 km, their average life span increase according ovserver on earth. But from moun frame it is not his time that dilated, but the distance to earth that become shorter. So with their average life span, they can cover a shorter distance compared the distance measured By observer on earth. So in this case, proper time is measured in moun frame but proper distance measured from earth frame. But for the case measuring speed like I mentioned in above equation, both the distance and time is proper according to one frame, and we need to calculated the same distance and time that being dilated and contracted By lorentz factor from other frame reference. So it is become multiplied By square of lorentz factor. (Note: this is verified By experiment)
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