How Does Time Dilation Affect Signal Transmission in a High-Speed Rocket?

The key is to use the right values for the variables and to be consistent. You did that. Good job!In summary, the light signal reaches the nose of the rocket clock in t = 3L/c according to both the rocket clock and Earth clocks. The rocket clock measures this time as t' = L/c, while Earth clocks measure it as t = 3L/[5(1-v/c)c], where L' is the contracted length of the rocket. Both methods give the same result.
  • #1
UrbanXrisis
1,196
1
a rocket ship of length L leaves Earth at a vertical speed of 4c/5. a light signal is sent vertically after it which arrives at the rocket tail at t=0 according to both rocket and Earth based clock. when does the signal reach the nose of the rocket clock according to (1) the rocket clockt (2) Earth clocks

(1)

t=L/c

(2)

[tex]t=\frac{L}{\sqrt{c^2-v^2}}[/tex]
[tex]t=\frac{L}{c\sqrt{1-(4/5)^2}}[/tex]
[tex]t=\frac{L}{\frac{3}{5} c}[/tex]

i know that this might be wrong but I'm not sure how to fix this mistake or which equation to use. any ideas?
 
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  • #2
UrbanXrisis said:
a rocket ship of length L leaves Earth at a vertical speed of 4c/5. a light signal is sent vertically after it which arrives at the rocket tail at t=0 according to both rocket and Earth based clock. when does the signal reach the nose of the rocket clock according to (1) the rocket clockt (2) Earth clocks

(1)

t=L/c

(2)

[tex]t=\frac{L}{\sqrt{c^2-v^2}}[/tex]
[tex]t=\frac{L}{c\sqrt{1-(4/5)^2}}[/tex]
[tex]t=\frac{L}{\frac{3}{5} c}[/tex]

i know that this might be wrong but I'm not sure how to fix this mistake or which equation to use. any ideas?
I don't think you have it. This is just the light pulse traveling at c chasing the nose of the ship traveling at 4c/5. The light pulse has to go farther by one ship length, but how long is the ship according to the Earth observer?
 
  • #3
[tex]\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}= \frac{1}{\sqrt{1-\frac{(3/5)^2 c^2}{u^2}}}= 5/4[/tex]

[tex]L'=\frac{4L}{5}[/tex]

so this means:

[tex]t=\frac{L'}{\sqrt{c^2-v^2}}[/tex]
[tex]t=\frac{L'}{c\sqrt{1-(4/5)^2}}[/tex]
[tex]t=\frac{\frac{4L}{5}}{{\frac{3}{5} c}}[/tex]
[tex]t=\frac{\frac{4L}{5}}{{\frac{3}{5} c}}[/tex]
[tex]t=\frac{4L}{3c}[/tex]

would this be the right answer?
 
  • #4
UrbanXrisis said:
[tex]\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}= \frac{1}{\sqrt{1-\frac{(3/5)^2 c^2}{u^2}}}= 5/4[/tex]

[tex]L'=\frac{4L}{5}[/tex]

so this means:

[tex]t=\frac{L'}{\sqrt{c^2-v^2}}[/tex]
[tex]t=\frac{L'}{c\sqrt{1-(4/5)^2}}[/tex]
[tex]t=\frac{\frac{4L}{5}}{{\frac{3}{5} c}}[/tex]
[tex]t=\frac{\frac{4L}{5}}{{\frac{3}{5} c}}[/tex]
[tex]t=\frac{4L}{3c}[/tex]

would this be the right answer?
It looks to me like you used the wrong velocity to calculate gamma, but you do need the contracted length. At t = 0 the light pulse is at the tail of the rocket of length L' (the contracted length that you need to fix). During the time t the light pulse will move ct and the nose of the ship will move vt. The light pulse must move a distance L' farther than than the nose. All the relativity transformations that you need are done when you calculate L'.
 
  • #5
could i used the equation:

[tex]t=\frac{t'+\frac{v x_1'}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}[/tex]

where:

t'=L/c
x_1=L
v=4c/5

I get an answer of t= 3L/c

does this seem correct?

also, the time measured from a clock in the spaceship would be just t'=L/c since the clock in the spaceship has a relative speed of 0 wrt the spaceship right?
 
Last edited:
  • #6
UrbanXrisis said:
could i used the equation:

[tex]t=\frac{t'+\frac{v x_1'}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}[/tex]

where:

t'=L/c
x_1=L
v=4c/5

I get an answer of t= 3L/c

does this seem correct?

also, the time measured from a clock in the spaceship would be just t'=L/c since the clock in the spaceship has a relative speed of 0 wrt the spaceship right?
t' = L/c is correct. Inside the spaceship it is just a pulse going from one end to the other with normal length L and normal time t'. Outside the ship it is

ct = L' + vt
(c - v)t = L'
(1-v/c)ct = L'
t=L'/[(1-v/c)c]
t = 3L/[5(1-v/c)c]
t = 3L/c

This is the same result you got by transforming the time as measured inside the ship to the time as measured outside the ship for two spatially separated events in S'. You can do it either way.
 

What is time dilation and how does it relate to a rocket?

Time dilation is a phenomenon in which time appears to run slower for objects that are moving at high speeds or are in strong gravitational fields. This is based on Einstein's theory of relativity. In the context of a rocket, time dilation refers to the difference in time experienced by an observer on the rocket compared to an observer on Earth.

How does the speed of a rocket affect time dilation?

According to the theory of relativity, as an object moves faster, time appears to pass slower for that object. This means that the faster a rocket travels, the more time dilation will occur. As the rocket approaches the speed of light, time dilation becomes more significant.

What is gravitational time dilation and how does it relate to a rocket?

Gravitational time dilation occurs when an object is in a strong gravitational field, such as near a planet or a black hole. This causes time to pass slower for the object in the gravitational field compared to an object in a weaker gravitational field. In the context of a rocket, this means that time will pass slower for the rocket as it approaches a planet or other massive object.

What is the twin paradox and how does it demonstrate time dilation in a rocket?

The twin paradox is a thought experiment that demonstrates the effects of time dilation. In this scenario, one twin stays on Earth while the other twin travels on a rocket at high speeds. When the traveling twin returns to Earth, they will have experienced less time than the twin who stayed on Earth. This is because time dilation occurred for the traveling twin due to their high velocity.

Is time dilation a proven phenomenon and has it been observed in real life?

Yes, time dilation has been proven through numerous experiments and has been observed in real-life scenarios. One example is the Global Positioning System (GPS), which relies on highly accurate time measurements. The clocks on GPS satellites must be adjusted to account for time dilation, as they are traveling at high speeds in orbit around Earth. Other experiments, such as the Hafele-Keating experiment, have also demonstrated the effects of time dilation.

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