You witnessed a hit-and-run accident when you were driving a car at velocity u t

Question

You witnessed a hit-and-run accident when you were driving a car at velocity u towards
a radio station very far ahead on the street. Here is how it happened: you were listening
to the radio when you saw" a car driving towards you hit a person standing by the bus
stop, everything happened in front of you and was closer than the radio station. The
accident happened when your favorite song from the radio just started, and after the
accident the car didn't change its velocity at all. You decided to stop the car by crashing
into him (without changing your velocity); and when your heroic crash stopped the car,
your favorite song from the radio just ended. Now the police wants you to help them
gure out the details of the accident.
As a warm up, let's do some multiple choice/True-or-False problems rst (2 pts each
problem):
According to another witness, code name witness A, who stood by the bus stop, the
victim struggled for a while before he died. Witness A doesn't have a watch, but he was
also listening to that radio and he reported that when the poor man died, it's just at the
end of that song you were listening.

1. In the rest frame, the victim died
a) before the car was stopped by you.
b) at the time the car was stopped by you.
c) after the car was stopped by you.

2. It's possible that in some frame the victim died after the car was stopped by you. (true or false)
3. According to witness A, after being hit the victim called his girlfriend but could
not nish his last words before he died. Due to time dilation, there exists some
other frame, where the poor man had more time before he died, and hence in
such a frame, the man nished his last words before he died. (true or false)

4.Because of the velocity addition formula, the speed reading by the man causing
the accident diers from the reading of his speed on the ground.

5. While from the point of view of witness A, the accident took place right at the
bus stop, due to Lorentz contraction the accident appeared to happen at a closer
place to you and hence it didn't happen at bus stop from your perspective.

6. The man claimed that due to Lorentz contraction although it looked like he would
hit the victim, there should actually be enough margin (in the direction perpen-
dicular to the direction he was moving) for him to pass the victim safely. Thus
in principle, he shouldn't be responsible for the accident.


7. Now, the police wants to know if the car was speeding. They are able to get
the measurement between the bus-stop and the wreckage of your car, and the distance
is L. But they don't have a record of the time t it took for the car to travel that
distance and calculate the speed. To help them, you realize that the length of your
favorite song from the radio station might help since you know the length of the song
is T when you usually listen to it from your CD player.
Find the speed v of the car. (Hint, don't forget that when you saw" the accident it's
actually already after the accident happened since it took time for the light to travel
from where it happened to your eye).


Thank you a lot !

Explanation / Answer

1. In the rest frame, the victim died a) before the car was stopped by you. 2.Its possible 3.True 4.False 5.True 6.False the change in length is very small order of (1-Sqrt(1-V^2/C^2)) V= velocity of car and C = Velocity of light 7.time in rest frame=T_0*gamma Time in rest frame=T_0=L/velocity of car in rest frame gamma=1/sqrt(1-v^2/C^2) ====>T_0/sqrt(1-v^2/C^2)=L/v solve the above we get velocity=L/sqrt(T^2+(L^2/C^2)) Source:http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/tdil.html

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