A question from a book about relativity

[m.medhat]

Homework Statement

Hello ,
I have a question please , I read in the book ( reflections on relativity ) that :-
Suppose a particle accelerates in such a way that it is subjected to a constant proper acceleration a0 for some period of time. The proper acceleration of a particle is defined as the acceleration with respect to the particle's momentarily co-moving inertial coordinates at any given instant. The particle's velocity is v = 0 at the time t = 0, when it is located at x = 0, and at some infinitesimal time t later its velocity is t a0 and its location is (1/2) a0 t2. The slope of its line of simultaneity is the inverse of the slope 1/v of its worldline, so its locus of simultaneity at t = t is the line given by
http://www.m5zn.com/uploads/2010/7/2/photo/0702100307469iwmjeb2nqrkvn4j.bmp
And my question is how did we derive the last equation ?

I need help please .

Homework Equations

The Attempt at a Solution

 

[vela]

That's just the point-slope form of a line:

y-y_0 = m(x-x_0)

where the line has slope m and passes through the point (x₀,y₀). In this case, you have t is in the role of y. Just plug in what the rest of the paragraph tells you and you'll get the derived formula.

 

[m.medhat]

very thanks .

 

[m.medhat]

Please I have another thing here , my book states that :-
“ This line intersects the particle's original locus of simultaneity at the point (x,0) “
I can’t understand this statement , please I want someone to explain and prove this statement for me .
I need help please .