Simple problem about relativity and train

[LCSphysicist]

Homework Statement

A train of proper length L moves at speed v1 with respect to the ground.
A passenger runs from the back of the train to the front at speed v2
with respect to the train. How much time does this take, as viewed by
someone on the ground?

Relevant Equations

...

I want to solve this question first using the train frame, and so going to the ground frame, but the things got wrong, so i would aprpeciate to know why.

I mean, there is another ways to solve it, but i want to know where is the error here.

Simply, in train frame the time elapsed between the event 'runs from the back' and 'coming in the front' is $$L/v2.$$

Now, why can't we go direct from this time to the ground interval? That is, $$L\gamma_{2}/(v2)$$

Actually, to get the right answer, we need to go first to person frame, and so to ground frame.

 

[PeroK]

Homework Statement:: A train of proper length L moves at speed v1 with respect to the ground.
A passenger runs from the back of the train to the front at speed v2
with respect to the train. How much time does this take, as viewed by
someone on the ground?
Relevant Equations:: ...

I want to solve this question first using the train frame, and so going to the ground frame, but the things got wrong, so i would aprpeciate to know why.

I mean, there is another ways to solve it, but i want to know where is the error here.

Simply, in train frame the time elapsed between the event 'runs from the back' and 'coming in the front' is $$L/v2.$$

Now, why can't we go direct from this time to the ground interval? That is, $$L\gamma_{2}/(v2)$$

Actually, to get the right answer, we need to go first to person frame, and so to ground frame.

When comparing the times of two events, the train frame and the ground frame are related according to the Lorentz Transformation (not simply time dilation). There is a combination of time dilation and the relativity of simultaneity.

 

[guv]

In addition to the formal Lorentz transformation approach, you could also use length and velocity observed in the ground frame and follow kinematics . Length can be found through length contraction and velocity can be found from relative velocity.