Adding Velocities
along Y, Zaxis
Let's complete our formulas for adding velocities with those related to Y and Zaxis, while assuming that the movement of one reference frame relatively to another is along the Xaxis.
Consider, as before, two inertial reference frames:
αframe with coordinates {X,Y,Z} and time T and
βframe with coordinates {x,y,z} and time t.
At time T=t=0 both reference frames coincide.
Assume that βframe uniformly moves relatively to αframe with speed v along αframe's Xaxis, maintaining parallelism of all corresponding axes.
Assume an object moves uniformly in βframe along some direction with X, Y and Zcomponents of its velocity vector u_{βx}, u_{βy} and u_{βz} correspondingly.
At time t=0 this object is at the origin of βframe.
Our task is to determine the components u_{αx}, u_{αy} and u_{αz} of the velocity of this object in αframe, using Lorentz Transformation.
In the previous lecture we discussed the analogous task but restricted only to an object moving in the βframe along Xaxis.
This lecture is about a more general movement of the object along any direction.
Obviously, the formula derived in the previous lecture for a movement along Xaxis is fully applicable to the Xcomponent of the general velocity vector, and out task is restricted to add Y and Zcomponents of the velocity vector.
We will use exactly the same methodology for Y and Zcomponents as we used for Xcomponent.
The object in βframe moves according to this formulas for its coordinates
x(t) = u_{βx}·t
y(t) = u_{βy}·t
z(t) = u_{βz}·t
Obviously,
dx(t)/dt = u_{βx}.
dy(t)/dt = u_{βy}.
dz(t)/dt = u_{βz}.
According to Lorentz transformation,
T = 


X = 


Y =  y  
Z =  z 
Our task is to find all the components of the velocity of an object in αframe, that is to find
u_{αx}=dX/dT
u_{αy}=dY/dT
u_{αz}=dZ/dT
All calculations we did in the previous lecture, when motion of an abject was restricted to Xdirection, are good for getting u_{αx}, but we will repeat them here.
Using an explicit form of functions x(t), y(t) and z(t), the above formulas of Lorentz Transformation are
T = 


X = 


Y =  u_{βy}·t  
Z =  u_{βz}·t 
Since
dX/dt = (dX/dT)·(dT/dt),
we can express the u_{αx}=dX/dT as
u_{αx} = dX/dT = (dX/dt)/(dT/dt)
and we can easily resolve our problem as follows
dX/dt = (v+u_{βx })/√1−(v/c)²
dT/dt = (1+v·u_{βx }/c²)/√1−(v/c)²
Therefore,
dX/dT = (v+u_{βx })/(1+v·u_{β }/c²)
The final formula for Xcomponent of the object's velocity u_{αx} in αframe is
u_{αx} = 

Let's analyze the Ycomponent of the object's velocity.
Since
dY/dt = (dY/dT)·(dT/dt),
we can express the u_{αy}=dY/dT as
u_{αy} = dY/dT = (dY/dt)/(dT/dt)
and we can easily resolve our problem as follows
dY/dt = dy/dt = u_{βy}
As before,
dT/dt = (1+v·u_{βx }/c²)/√1−(v/c)²
Therefore,
u_{αy} = 

In particular, if an object moves along yaxis in βframe (u_{βx}=0), its Yaxis speed in αframe is
u_{αy} = u_{βy}·√1−(v/c)²
The Zcomponent of the velocity is analogous to Ycomponent.
Since
dZ/dt = (dZ/dT)·(dT/dt),
we can express the u_{αz}=dZ/dT as
u_{αz} = dZ/dT = (dZ/dt)/(dT/dt)
and we can easily resolve our problem as follows
dZ/dt = dz/dt = u_{βz}
As before,
dT/dt = (1+v·u_{βx }/c²)/√1−(v/c)²
Therefore,
u_{αz} = 

In particular, if an object moves along zaxis in βframe (u_{βx}=0), its Zaxis speed in αframe is
u_{αz} = u_{βz}·√1−(v/c)²
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