How can I show that z = v/c for small velocities

[Logarythmic]

How can I show that

$$1+z=\sqrt{\frac{1+v/c}{1-v/c}}$$

becomes $$z \simeq v/c$$ for small velocities? Please give me a hint.

 

[cristo]

Try writing the right hand side as $$(1+v/c)^{1/2}(1-v/c)^{-1/2}$$. Can you expand this?

 

[Logarythmic]

So

$$1+z=\frac{\sqrt{1+\beta}}{\sqrt{1-\beta}}\simeq \left( 1+\frac{1}{2}\beta-\frac{1}{8}\beta^2+... \right) \left( 1+\frac{1}{2}\beta+\frac{3}{8}\beta^2+... \right) = 1+\beta+\frac{5}{8}\beta^2+...=1+\frac{v}{c}$$

Correct?

 

[cristo]

Yup, that's correct.

 

[sara_87]

cristo scroll up, lol

 

[Gib Z]

I would think there was an easier way...v is approaching zero, sub v=0 into the right hand side and its easy to see z is also approaching zero...

PS: Cristo has 666 posts...ooh