Length Contraction: Sally & Sheldon's Meter Stick

In summary, Sally claims that Sheldon's meter stick is 0.8 m long when in fact it is only 0.53 m long due to relativistic velocity addition.
  • #1
Lucille
31
0

Homework Statement


Sally and Sheldon have identical meter sticks. Sally is on earth, and sheldon is in a spacecraft that moves at 0.5c relative to sally. Sheldon leaves the spaceship at 0.1c relative to the spacecraft in a launch pad, moving away from sally. according to sally, what is the length of sheldon's meter stick?

Homework Equations


x=gamma*(x-vt)

The Attempt at a Solution


if in sheldon's frame:
Lo = 1m = length of meter stick
In sally's frame:
L=gamma*(x'-vt'-x-vt)
but t1=t2

so L=Lo/gamma
L= 0.8 m

So Sally sees sheldon's meter stick as 0.8 m long.
 
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  • #2
You have not defined t1 and t2.
What did you use for v?
 
  • #3
I used v=0.6c

But as I was reviewing my notes, I found that
L=Lo/gamma
where Lo is in Sheldon's' frame and L is in Sally's frame - both of Sheldon's meter stick

So would it be:
L=Lo/(1/(sqrt(1-v^2/c^2)
=Lo*sqrt(1-v^2/c^2)
=(1m)*(sqrt 1-(0.6^2))
=0.8 m
 
  • #4
So for the velocity you have used v = 0.5c+0.1c = 0.6c. This is not correct in relativity and you need to apply relativistic velocity addition. Otherwise, it would be correct to use the length contraction formula as you have done, you just need to use relativistic velocity addition to find the velocity to put in.
 
  • #5
Would the velocity of the launch craft in the (x,ct) frame be:

u = (u' + v)/(1+u'v/c^2)

where u = velocity of launch craft in x,ct
v = velocity of the spacecraft in x,ct
and u' is the velocity of the launch craft in x,ct

so u = 0.4c?

Then L = gamma*Lo = (1m)*(sqrt(1-0.4^2)) = 0.917m?
 
  • #6
What numbers did you put to get 0.4c? The launch pad is moving faster than the spacecraft and should have a velociy of at least 0.5c.
 
  • #7
I used u'= 0.1c v=0.5c

But if u' should be at least 0.5c, would it be u'=0.5c+0.1c = 0.6 c?

So:

u=(u' + v)/(1+u'v/c^2)
where u' = 0.6c, v=0.5c

u= 0.85c?

L=0.53m??
 
  • #8
Wait,

I may have substituted in the wrong numbers in the first place

u = (u' + v)/(1-u'v/c^2)
= (0.1c+0.5c)/(1-(0.1c)(0.5c)/c^2)
=0.57c

L=0.82m?
 
  • #9
Sorry, I was a bit sloppy, the launchpad should have a velocity of at least 0.5c in the Earth frame.

You should be using what you said first (u'=0.1c and v=0.5c), but it should not evaluate to u=0.4c.
 
  • #10
Lucille said:
Wait,

I may have substituted in the wrong numbers in the first place

u = (u' + v)/(1-u'v/c^2)
= (0.1c+0.5c)/(1-(0.1c)(0.5c)/c^2)
=0.57c

L=0.82m?

This looks more in line with what I would expect apart from that there should be a + in the denominator (although your final numbers for u and L are correct so you probably used it like as the - was a +).
 
  • #11
Oops, yes, that was supposed to be a plus.

Thank you so much! :)
 
  • #12
Well done and thanks Orodruin.
 

1. What is length contraction?

Length contraction is a phenomenon in which the length of an object appears to contract when it is in motion relative to an observer.

2. How does length contraction occur?

Length contraction occurs due to the effects of special relativity, which states that time and space are relative to the observer's frame of reference. As an object moves at high speeds, its length in the direction of motion appears to decrease from the perspective of an observer at rest.

3. What is the relationship between velocity and length contraction?

The relationship between velocity and length contraction is that the faster an object moves, the greater the amount of length contraction will occur. This means that the length of an object will appear shorter to an observer as its velocity increases.

4. Can length contraction be observed in everyday life?

No, length contraction is only observable at very high speeds, close to the speed of light. In everyday life, the effects of length contraction are too small to be noticed.

5. How does "Sally & Sheldon's Meter Stick" demonstrate length contraction?

"Sally & Sheldon's Meter Stick" is a thought experiment that involves two observers, Sally and Sheldon, who are moving at different speeds relative to each other. They both measure the length of a meter stick that is moving with them, and they will get different results due to the effects of length contraction. This illustrates the concept of length contraction in a simple and relatable way.

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