How Does Particle Velocity Change in Different Reference Frames?

[CINA]

Homework Statement

A particle moves at speed $$u_{x}= +0.0c$$ according to an observer in the frame S.

a)Calculate the velocity of the particle, $$u_{x}'_{Gal}$$ and $$u_{x}'_{SR}$$ as measured by an observer in S'. Let $$\beta$$ range from -0.95c to +0.95c in increments of 0.05. Put your results in a table.

b)Graph $$u_{x}'_{Gal}$$ and $$u_{x}'_{SR}$$ as a function of $$\beta$$.

c) Repeat steps a and b for particles with $$u_{x}=+0.5c$$, $$u_{x}=+0.8c$$, $$u_{x}=+0.99c$$ and $$u_{x}=+0.9999c$$.

*Note* $$u_{x}'_{Gal}$$ and $$u_{x}'_{SR}$$ mean the velocities with the Galilean and the lorentz trans., respectfully.

Homework Equations

$$u'=\frac{u+v}{1+\frac{(u)(v)}{c^{2}}}$$

The Attempt at a Solution

These are kind of dumb questions but here goes:

First of all the equation with beta in it looks like:

$$u'=\frac{u+v}{1+\frac{(u)(\beta)}{c}}$$

Right?

Secondly, when ux=+0.0c the equation just reduces to u'x=v correct?

Thirdly, when it says graph u'Gal and u'SR as functions of beta that means beta on the horizonal and the velocities on the vertical right?

 

[vela]

Homework Statement

A particle moves at speed $$u_{x}= +0.0c$$ according to an observer in the frame S.

a)Calculate the velocity of the particle, $$u_{x}'_{Gal}$$ and $$u_{x}'_{SR}$$ as measured by an observer in S'. Let $$\beta$$ range from -0.95c to +0.95c in increments of 0.05. Put your results in a table.

b)Graph $$u_{x}'_{Gal}$$ and $$u_{x}'_{SR}$$ as a function of $$\beta$$.

c) Repeat steps a and b for particles with $$u_{x}=+0.5c$$, $$u_{x}=+0.8c$$, $$u_{x}=+0.99c$$ and $$u_{x}=+0.9999c$$.

*Note* $$u_{x}'_{Gal}$$ and $$u_{x}'_{SR}$$ mean the velocities with the Galilean and the lorentz trans., respectfully.

Homework Equations

$$u'=\frac{u+v}{1+\frac{(u)(v)}{c^{2}}}$$

The Attempt at a Solution

These are kind of dumb questions but here goes:

First of all the equation with beta in it looks like:

$$u'=\frac{u+v}{1+\frac{(u)(\beta)}{c}}$$

Right?

First, what is β defined as? The velocity of what relative to what? Second, in the problem statement you gave, β is proportional to c, so it's a regular velocity. Usually β denotes a velocity as a fraction of the speed of light, i.e. β=0.5 not β=0.5c. If you're using this convention, then yes, what you wrote above is correct because one of the factors of c in the formula was pulled into β.

Secondly, when ux=+0.0c the equation just reduces to u'x=v correct?

Yes.

Thirdly, when it says graph u'Gal and u'SR as functions of beta that means beta on the horizonal and the velocities on the vertical right?

Yes.