How Does Velocity Addition in Special Relativity Ensure u' Remains Less Than c?

[byerly100]

Consider two reference frames, S and S', moving with speed v (<c) with respect to one another along the x direction.

If a certain object moves with velocity u (u<c) with respect to S, and velocity u' with respect to S', use the velocity addition equations (in three dimensions) to show that u'<c.

 

[quasar987]

What have you tried?

 

[Meir Achuz]

Consider two reference frames, S and S', moving with speed v (<c) with respect to one another along the x direction.

If a certain object moves with velocity u (u<c) with respect to S, and velocity u' with respect to S', use the velocity addition equations (in three dimensions) to show that u'<c.

Just write down the equations and substitute.
Then find u"^2.

 

[byerly100]

Are you saying to take the derivative of u'?

 

[Meir Achuz]

Don't you know the vdlocity addition eqs in SR?
If not you have to find the ratio dx'/dt' and dy'/dt' in terms of the unprimed using the Lorentz transformation eqs. Just take the ratios. Differentiation is not needed. If the know the eqs., just substitute.