The Classic Pole in Barn Relativity Question

[little neutrino]

Homework Statement

A pole-vaulter holds a 5.0 m pole. A barn has doors at both ends, 3.0 m apart. The pole-vaulter on the outside of the barn begins running toward one of the open doors, holding the pole level in the direction he is running. When passing through the barn, the pole just fits entirely within the barn all at once.

According to the pole-vaulter, which occurs first, the front end of the pole leaving the barn or the back end entering? Explain. What is the time interval between these two events according to the pole-vaulter?

Homework Equations

t' = γ(t - ux/c²)

The Attempt at a Solution

Let S be reference frame of stationary observer
Let S' be reference frame of pole-vaulter
Subscript 1: Front end of pole leaving barn
Subscript 2: Back end of pole entering barn

From stationary observer's POV, back end of pole enters barn at the same time as front end of pole leaves barn. (Is this inference correct??)
t₁ - t₂ = 0

t₂' - t₁' = ... = γ[(t₁ - t₂) + 3u/c²]
Since t₂' - t₁' > 0, t₁' occurs first. Therefore, front end of pole leaves barn first.

Is the calculation correct? Is there a more intuitive way of understanding which comes first?

Thanks!

 

[stevendaryl]

Homework Statement

A pole-vaulter holds a 5.0 m pole. A barn has doors at both ends, 3.0 m apart. The pole-vaulter on the outside of the barn begins running toward one of the open doors, holding the pole level in the direction he is running. When passing through the barn, the pole just fits entirely within the barn all at once.

According to the pole-vaulter, which occurs first, the front end of the pole leaving the barn or the back end entering? Explain. What is the time interval between these two events according to the pole-vaulter?

Homework Equations

t' = γ(t - ux/c²)

The Attempt at a Solution

Let S be reference frame of stationary observer
Let S' be reference frame of pole-vaulter
Subscript 1: Front end of pole leaving barn
Subscript 2: Back end of pole entering barn

From stationary observer's POV, back end of pole enters barn at the same time as front end of pole leaves barn. (Is this inference correct??)
t₁ - t₂ = 0

t₂' - t₁' = ... = γ[(t₁ - t₂) + 3u/c²]
Since t₂' - t₁' > 0, t₁' occurs first. Therefore, front end of pole leaves barn first.

Is the calculation correct? Is there a more intuitive way of understanding which comes first?

Thanks!

Yes, your calculations are correct. The intuitive way of understanding it might be to look from the point of view of the pole-vaulter. From his point of view, it's the barn that is moving, and is length-contracted to just $9/5$ meters between the doors. (You've described a length contraction factor, $\frac{1}{\gamma}$, of $3/5$). So if you stick a 5-meter pole into a barn that is only 9/5 meters long, then of course the front end of the pole will come out the back door before the back end comes through the front door.

 

[little neutrino]

Yes, your calculations are correct. The intuitive way of understanding it might be to look from the point of view of the pole-vaulter. From his point of view, it's the barn that is moving, and is length-contracted to just $9/5$ meters between the doors. (You've described a length contraction factor, $\frac{1}{\gamma}$, of $3/5$). So if you stick a 5-meter pole into a barn that is only 9/5 meters long, then of course the front end of the pole will come out the back door before the back end comes through the front door.

Ohhh right! Thanks! :)