Simple Special Relativity Homework: Time Dilation and Length Contraction

[hotcommodity]

Homework Statement

Anna is on a railroad flatcar moving at 0.6c relative to Bob. Their clocks read 0 as Anna's center of mass passes Bob's. Anna's arm is outstretched in the direction the flatcar moves, and in her hand is a flashbulb. According to the wristwatch on Anna's hand, the flashbulb goes off at 100ns. The time of this event differs by 27ns. (a) Is it earlier or later than 100ns? (b) How long is Anna's arm (from hand to center of mass) ?

Homework Equations

Transformation Eq.'s:

1. x' = \gamma(x - vt)
2. t' = \gamma(-vx/c² + t)
3. x = \gamma(x' + vt')
4. t = \gamma(vx'/c² + t')

5. \gamma = 1/sqrt(1 - v²/c²)

The Attempt at a Solution

For part a, I found gamma to be 1.25, and the time in Bob's frame to be 125ns (not differing by 27ns as the book suggested).

For part b, which I've tried many times, I get an answer of 0m (i.e. the bulb flashes at Anna's center of mass). I call Anna's center of mass at the time of the bulb flash Event 1, and the flash of the bulb Event 2. I also use the convention of Anna's frame of reference as being primed (e.g. x', t', and so on).These are the quantities I have, maybe someone can spot an error in my reasoning:

Event 1: t1' = 100ns, x1' = 0, t1 = ?, x1 = ?

Event 2: t2' = 100ns, x2' = ?, t2 = 125ns (from part a), x2 = 22.5 m (d=vt --> x2 = .6c(125ns) = 22.5m)

Using the transformation eq.'s, I get x1 = 0, which gives me zero for the uncontracted length. Any help is appreciated.

 

[tiny-tim]

Anna is on a railroad flatcar moving at 0.6c relative to Bob. Their clocks read 0 as Anna's center of mass passes Bob's. Anna's arm is outstretched in the direction the flatcar moves, and in her hand is a flashbulb. According to the wristwatch on Anna's hand, the flashbulb goes off at 100ns. The time of this event differs by 27ns. (a) Is it earlier or later than 100ns? (b) How long is Anna's arm (from hand to center of mass) ?

For part a, I found gamma to be 1.25, and the time in Bob's frame to be 125ns (not differing by 27ns as the book suggested).

Hi hotcommodity!

No … you're misunderstanding the question.

You've worked out Bob's measurement of time between two things at the same place (as measured by Alice).

The question asks for Bob's measurement of time between Alice passing him and the flashbulb (in her hand) going off.

Try again!

 

[hotcommodity]

Hi hotcommodity!

No … you're misunderstanding the question.

You've worked out Bob's measurement of time between two things at the same place (as measured by Alice).

The question asks for Bob's measurement of time between Alice passing him and the flashbulb (in her hand) going off.

Try again!

I'm not sure I understand. Are you saying I misunderstood the question being asked for part a or part b?..

 

[tiny-tim]

I'm not sure I understand. Are you saying I misunderstood the question being asked for part a or part b?..

part a.