Chasing a Beam of Light: Understanding Special Relativity's Thought Experiment

[QuantumKing]

Ok, so I read about a thought experiment pertaining to special relativity in a book somewhere and the experiment was that of chasing a light beam, which I am sure everyone here has heard of. For those who haven't heard of such an "experiment" here is how it goes:

Bart has just bought himself a new nuclear powered skateboard able to reach a speed of 260,000km/s, he decides to have a race against a beam of light using his skateboard. His sister Lisa has a laser beam and stands at rest on a flat never ending plane. Both have sychronized atomic clocks, before Bart starts moving. So Lisa starts the countdown and then they're off! From Lisa's perspective, Bart is being outrun by the light beam by only 40,000km/s, but from Bart's perspective he is being untrun by the speed of light, 300,000km/s.

Now, i can't seem to make sense of this, if you look at the equations explaining time dilationand length contraction, the gamma factor is canceled out when you divide length by time in any reference frame. So if i were to say that from Bart's perspective the light beam is 300,000km ahead of him after he measures 1 second on his clock, and from Lisa's persepective the light beam (using time dilation and length contraction equations) would be 600,000km ahead of him after 2 seconds (I used 260,000km/s for a reason), meaning Bart is being outrun by the speed of light in her reference frame as well. This is where I am lost and can't make sense of that statement above where Lisa's persepective shows Bart being outrun by only 40,000km/s. I'm obviously not looking at it the right way. Anyone have an idea?

 

[jtbell]

Velocities don't "add" in the same way in relativity as they do in classical physics.

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/einvel.html

I don't think you can derive the relativistic velocity addition formula just from length contraction and time dilation, because they don't give a complete description of relativistic kinematics. You have to include relativity of simultaneity also.

However, all of these things can be derived from the Lorentz transformation

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/ltrans.html#c2

which in turn can be derived from the fundamental principles of relativity (speed of light is constant, etc.)

 

[Wildreamz]

I'm not an expert in this subject, but this is my point of view:

Lisa's Frame of Ref
Lisa is stationary.
The moment Bart start moving is simultaneous with Lisa's shining of torch.
@time lapsed=1s
Distance traveled by Bart=260,000km
Distance traveled by the first photon=300,000km
Bart is outran by 40,000km.

Bart's Frame of Ref
Bart is staionary.
Lisa is moving away in the opposite direction at the speed of 260,000km/s.
Due to time dilation, time lapsed for Bart=1/γs (start of his motion till he is outran by 40,000km)

γ=(1-2.6^2/3^2)^-0.5=2.00

Time lapsed=1/γs=0.5s
Taking displacement by Lisa from Bart to be +ve:
Displacement of Lisa= 260,000*0.5km = 130,000km
Displacement of photon=-40,000/γkm=-20,000km

Hmm.. How come the photon did not pass Bart by 150,000km after 0.5s?
If we take Lisa's reference frame, the moment Bart start his motion at 260,000km/s is simultaneous with the moment Lisa shines her torch and start her clock(3 simultaneous events). However, in Bart's reference frame, he started his clock at the same moment he started motion, but Lisa only shone her light after some time = w.

Bart's frame of Ref:
Displacement of Photon(from Bart)=Displacement of Lisa + Displacement of Photon(from Lisa)=-20,000km
w*260,000-(0.5-w)300,000=-20,000
560,000w=130,000
w=13/56s=0.232s (approx.)

@t=0.232s
Distance traveled by Lisa=260,000*13/56km=60357km
She shone her light.
Time traveled by the first photon of her torch 0.5-13/56s=15/56s
Displacement of the first photon of her torch(from the point she shine her torch)=-300,000*15/56s=-80357km
Therefore distance in which the photon outrun Bart=60357-80357km=-20,000km.

In his frame of ref, Bart is stationary, he is completely "outran" by light by 300,000km/s.

Are my calculations and reasonings correct this time?

 

[joyful55]

From what I undestand of relativity is that the only thing that is constant in all frames of reference is the speed of light. As for distance, there is always length contraction, hence, the distance can never be simply added up using simple classical mechanics. And once you apply Lorentz transformation, you're going to define which fram are you measuing from and thus where is the origin in that frame.