Time dilation and length contraction problem

[squishy-fish]

Homework Statement

Anna is on a railroad flatcar moving at 0.6c relative to Bob. Her arm is outstretched in the dirction the flatcar moves, and in her hand is a flashbulb. According to her wristwatch, the bulb goes off at 100 ns. This even according to Bob differs by 27 ns.
a) is it earlier or later than 100 ns?
b) how long is Anna's arm?

Homework Equations

lambda_v = 1 / sqrt[(1-v²) / c²]

The Attempt at a Solution

a) I'm a little bit confused by time dilation. I think that since Anna is moving according to Bob's frame of reference, he sees a longer time interval for her. Therfore, his watch would read later than 100 ns. I think?


b)lambda_v = 1 / sqrt[(1-v²) / c²]
= sqrt(1-0.6²)

=1.25

I'm confused about where to go from here. I need the length of Anna's arm. I know that this is the length from her hand to her center of mass. So if she is moving at 0.6c relative to Bob, and he sees the light go off at 127 ns,
d = vt
= 0.6c(127)
=76.2 cm

L₀ = L x lambda_v
= 76.2 x 1.25
= 95.25 cm

So I found the length of her arm to be 95 cm. However, I have a feeling this isn't correct. I'm not sure I used the right formula...

 

[Modulated]

Homework Statement

Anna is on a railroad flatcar moving at 0.6c relative to Bob. Her arm is outstretched in the dirction the flatcar moves, and in her hand is a flashbulb. According to her wristwatch, the bulb goes off at 100 ns. This event according to Bob differs by 27 ns.

OK, this is slightly unclear at the moment but I think the intention is that Anna ('s body) passes Bob at t = 0. This will become important!

Homework Equations

lambda_v = 1 / sqrt[(1-v²) / c²]

OK, not a bad start (although I'd call this quantity $\gamma$). Any other equations that might be relevant?

The Attempt at a Solution

a) I'm a little bit confused by time dilation. I think that since Anna is moving according to Bob's frame of reference, he sees a longer time interval for her. Therfore, his watch would read later than 100 ns. I think?

Yep, that's right.

b)lambda_v = 1 / sqrt[(1-v²) / c²]
= sqrt(1-0.6²)

=1.25

So far so good.

I'm confused about where to go from here. I need the length of Anna's arm. I know that this is the length from her hand to her center of mass. So if she is moving at 0.6c relative to Bob, and he sees the light go off at 127 ns,
d = vt
= 0.6c(127)
=76.2 cm

Not quite. 0.6c is Anna's velocity; 127 ns is the time interval in Bob's reference frame between her passing him and the flash going off. So you can't multiply these numbers together directly like this. Hence why I asked whether you knew any further equations that might be useful. Later on in your post you used the (correct) formula $L' = L\gamma$ for the transformation of a length interval; do you know similar formulas for transforming a position or time coordinate?

 

[squishy-fish]

Okay. So does that mean that I should use the formula t = gamma_v x t₀ ?

Then I could plug in 100 x 10^-6 for t₀ since this is how long it took in anna's frame of reference?
This gives me 1.25 x 10^-5 as my t. and then can I use this t instead to find length?
But I still get a very similar answer when I do this.

 

[Modulated]

Okay. So does that mean that I should use the formula t = gamma_v x t₀ ?

This is also for a time interval. Do you understand what I mean by the difference between a coordinate and an interval? And do you recognise the formula, say,

$t' = \gamma\left(t - \dfrac{vx}{c^2}\right)$ ?

 

[rwisz]

I would like to clarify a few things about time dilation and length contraction in this scenario. Time dilation is a phenomenon in which time appears to pass slower for an object in motion relative to an observer. This means that Anna's wristwatch, which is moving at 0.6c, will appear to tick slower for Bob who is at rest.

In this case, when Anna's wristwatch reads 100 ns, Bob's wristwatch will read a later time of 127 ns. This is because Bob is observing Anna's wristwatch from a different frame of reference, which is moving relative to him. Therefore, it is later than 100 ns according to Bob's frame of reference.

As for the length contraction, it is a phenomenon in which an object appears shorter in the direction of motion when observed from a different frame of reference. In this case, Anna's arm will appear shorter to Bob when she is moving at 0.6c. The formula you have used for length contraction is correct, but the value you have calculated for the length of Anna's arm is not accurate. This is because the formula you have used is for calculating the length of an object in motion, not the length of a moving object's arm.

To calculate the length of Anna's arm, we need to use the formula L = L0 / lambdav, where L0 is the rest length of her arm and lambdav is the Lorentz factor. Since we do not know the rest length of Anna's arm, we cannot accurately calculate the length of her arm in this scenario. However, we can say that her arm will appear shorter to Bob due to length contraction.

In conclusion, the flashbulb going off at 100 ns according to Anna's wristwatch will appear to happen later than 100 ns according to Bob's wristwatch. Additionally, Anna's arm will appear shorter to Bob due to length contraction. I hope this clarifies the concepts of time dilation and length contraction in this scenario.