Calculating Velocity Components in Moving Frames

[Abdul.119]

Homework Statement

Consider a frame that is moving with velocity v in the positive x-direction. An observer in this moving frame measures the velocity of the particle in the x- and z-direction, u_x' = 0.9c, u_z'=0
a) What is the maximum velocity of the particle in the y-direction u_y' measured by the observer in the moving frame?
b) What velocity components does an observer in a frame at rest measure in the x-, y- and z-direction for v = 0.5c

Homework Equations

v(t) = v_x(t) + v_y(t) + v_z(t)
u_x' = u_x - v / (1-(v/c^2)
u_y' = u_y / (gamma_v (1-(v/c^2) * u_x)
u_z' = u_z / (gamma_v (1-(v/c^2) * u_x)
gamma_v = 1 / sqrt(1-(v/c^2))

The Attempt at a Solution

I am not sure but I thought the first equation can be used, so it would be 0.9 + 0 + u_y' = 1, then solve for u_y' ? so u_y' = 0.1c ??
and for part b) , I use the values of u_x', u_y' and u_z', with the given value of v = 0.5c in the equations, to solve for u_x, u_y and u_z, but I'm not sure do I plug in the v as 0.5c, or use the actual speed of light 0.5*3*10^8 ?

 

[mfb]

Are you sure all indices in the problem statement are right?
If the observer is moving in x-direction and sees the particle moving in x-direction only, the particle moves in the x-direction only for the lab frame as well, which makes most of the questions meaningless.

 

[Abdul.119]

Are you sure all indices in the problem statement are right?
If the observer is moving in x-direction and sees the particle moving in x-direction only, the particle moves in the x-direction only for the lab frame as well, which makes most of the questions meaningless.

These are the equations I had in my notes, so I don't know, how is the problem solved then?

Edit: Yes, I just checked my textbook and these equations are correct (at least the last 4 equations).

 

[TSny]

From the measured values of the x' and z' components of velocity, we are supposed to determine the maximum possible value for the y' component of velocity. At least that's how I interpret the problem.

Abdul, you will need to recall how the speed of a particle is determined from the components of velocity. (The speed is not the sum of the velocity components.)

Your second "relevant equation" is incorrect as written.

Yes, you can just substitute 0.5c for v without needing to enter the value of c.

 

[Abdul.119]

From the measured values of the x' and z' components of velocity, we are supposed to determine the maximum possible value for the y' component of velocity. At least that's how I interpret the problem.

Abdul, you will need to recall how the speed of a particle is determined from the components of velocity. (The speed is not the sum of the velocity components.)

Your second "relevant equation" is incorrect as written.

Yes, you can just substitute 0.5c for v without needing to enter the value of c.

So the equations are not relevant? what should I use then?

 

[TSny]

Your first relevant equation is not right at all. Speed, v, is the magnitude of the velocity vector. How do you calculate the magnitude of a vector from the components of the vector?

The rest of your relevant equations are very relevant. However, you did not type the equation for u_x' correctly.

 

[Abdul.119]

Your first relevant equation is not right at all. Speed, v, is the magnitude of the velocity vector. How do you calculate the magnitude of a vector from the components of the vector?

The rest of your relevant equations are very relevant. However, you did not type the equation for u_x' correctly.

Oh yes, sorry there is a u_x missing in the right hand side of that equation

So the magnitude of the velocity vector is V = sqrt [Vx^2 + Vy^2 + Vz^2] , I use that to find the velocity in the y direction?

 

[TSny]

Yes.

 

[Abdul.119]

Yes.

And what does the V equal? 1? so that 1 = sqrt [0.9^2 + Vy^2 + 0] ?

 

[TSny]

Yes, if you are taking c = 1.

 

[Abdul.119]

Yes, if you are taking c = 1.

Thanks for the help, the solution makes sense to me now