Special Relativity -- Velocity transformation

[Lito]

Homework Statement

A train travels in the +x direction with a speed of β = 0.80 with respect to the ground. At a certain time, two balls are ejected, one traveling in the +x direction with x-velocity of +0.60 with respect to the train and the other traveling in the −x direction with x-velocity of −0.40 with respect to the train.

(a) What are the x-velocities of the balls with respect to the ground?

(b) What is the x-velocity of the first ball with respect to the second? [Hint: The frames you will choose to be the Home Frame and the Other Frame for part (a) will not be the same as your choices for part (b).]

Homework Equations

The Attempt at a Solution

(a) Using the transformation equations: β= 0.8

Train Frame:
$$ V'_{xL}= +0.6 , V'_{xR}= -0.4 $$
Ground frame:
$$ V_{xR}=\frac{0.6+0.8}{1+0.6*0.8} = 0.94 $$
$$ V_{xL}=\frac{-0.4+0.8}{1-0.4*0.8}= 0.58 $$

** No question about this section **

(b)
I thought taking the Train frame as the home frame and the +0.6 ball as the Other frame.Drawing the -0.4 ball worldline, I can take a random event on this line and transform Δx and Δt in the home frame to Δx' and Δt' in the Other frame.

Calculating Δx' /Δt' will evaluate the speed of the -0.4 ball with respect to +0.6 ball.Is this process correct?

Thanks so much!

 

[SammyS]

Homework Statement

A train travels in the +x direction with a speed of β = 0.80 with respect to the ground. At a certain time, two balls are ejected, one traveling in the +x direction with x-velocity of +0.60 with respect to the train and the other traveling in the −x direction with x-velocity of −0.40 with respect to the train.

(a) What are the x-velocities of the balls with respect to the ground?

(b) What is the x-velocity of the first ball with respect to the second? [Hint: The frames you will choose to be the Home Frame and the Other Frame for part (a) will not be the same as your choices for part (b).]

Homework Equations

The Attempt at a Solution

(a) Using the transformation equations:β= 0.8

Train Frame:
$$ V'_{xL}= +0.6 , V'_{xR}= -0.4 $$
Ground frame:
$$ V_{xR}=\frac{0.6+0.8}{1+0.6*0.8} = 0.94 $$
$$ V_{xL}=\frac{-0.4+0.8}{1-0.4*0.8}= 0.58 $$

** No question about this section **

(b)
I thought taking the Train frame as the home frame and the +0.6 ball as the Other frame.Drawing the -0.4 ball worldline, I can take a random event on this line and transform Δx and Δt in the home frame to Δx' and Δt' in the Other frame.

Calculating Δx' /Δt' will evaluate the speed of the -0.4 ball with respect to +0.6 ball.Is this process correct?

Thanks so much!

(Try not to use so much Bold face type.)

I'm uncertain whether that process is correct or not.

I would be inclined to use one of the balls ( probably the -0.4 ball) as one of your frames and the train as the other.

You know the velocity of the train in the frame of the ball, and the velocity of the other ball in the frame of the train.