# Conservation of linear momentum at relativistic speeds

#### carl fischbach

The question I ask is linear momentum conserved in
in instance cited below?

You place a particle at the origin on a x-y axis
and accelerate it to 61% of c in the y direction.
Then you accelerate it to 61% of c in the x direction.
The net velocity of the particle will be
86% of c at 45 degrees.The key here is that it
takes approximately 3 times the energy to
accelerate the particle in the x direction than
the y direction, due to the fact that the net
velocity change in the y direction is 0%-61% of c
and in x direction the net velocity change is
61%-86% of c.If the rate of acceleration,distance
of acceleration and time of acceleration are the
same on the x and y axis, then force of acceleration on
the x axis has to be greater than
on the y axis, since the energy of acceleration
on the x axis is 3 times that of the y axis.
Therefore the momentum on the x axis is greater
the y axis.

If the particle's final velocity is 86% of c at
45 degrees then the momentum of acceleration
should be equal on both the x and y axis.
Is there a discrepancy in momentum here?

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#### Labguy

Originally posted by carl fischbach
The question I ask is linear momentum conserved in
in instance cited below?

You place a particle at the origin on a x-y axis
and accelerate it to 61% of c in the y direction.
Then you accelerate it to 61% of c in the x direction.
The net velocity of the particle will be
86% of c at 45 degrees.The key here is that it
takes approximately 3 times the energy to
accelerate the particle in the x direction than
the y direction, due to the fact that the net
velocity change in the y direction is 0%-61% of c
and in x direction the net velocity change is
61%-86% of c.If the rate of acceleration,distance
of acceleration and time of acceleration are the
same on the x and y axis, then force of acceleration on
the x axis has to be greater than
on the y axis, since the energy of acceleration
on the x axis is 3 times that of the y axis.
Therefore the momentum on the x axis is greater
the y axis.

If the particle's final velocity is 86% of c at
45 degrees then the momentum of acceleration
should be equal on both the x and y axis.
Is there a discrepancy in momentum here?
I don't think so. No acceleration above 0.61c is used or provides momentum. The 0.86c is the apparent VECTORED V (in your example). But, I don't think that your example is correct since you seem to be using two-dimensional (graph-paper) trig here, while with excluding time. In V calculations, a Y velocity at 0.61c and an X velocity at 0.61c takes time to reach the "end-point" from where you measure the Hypotenuse. The "actual V" of the vectored triangle cannot exceed the V of the greater of the XV or the YV if time is included. The "apparent" V can though, as is seen in apparent superluminal expansion around some supernova remnants.

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