# Estimating the lorentz factor of a spaceship

## Homework Statement

ESTIMATE the Lorentz factor of a spaceship which travels to a nearby star 10pc from the solar system in 50 years spaceship time.

## Homework Equations

Lorentz factor: γ = 1/√(1 - v^2/c^2 )

Time dilation: Δt' = γΔt

## The Attempt at a Solution

Using the distance and time given, from the spaceship's point of view, it is travelling at a velocity of 0.65c.

I think I need to figure out the velocity from the point of view of Earth and then put that into the lorentz factor, however I don't know how to do that and that doesn't seem like an estimate to me.

I also thought that perhaps if I could work out the time it took to get there from the point of view of Earth I could rearrange the time dilation equation and get gamma from that.

I think my problem is I don't know what I can reasonably assume/estimate without getting this completly wrong.

Help :(

## Answers and Replies

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Simon Bridge
Homework Helper
It takes 50years spaceship time to go 10pc in the Earth frame.
10pc is about 350ly so that is definately relativistic.

estimate the relative speed by v=d/T=d'/T' (that's average speed right?)
observers on the spaceship and Earth agree about v and disagree about who is moving.

how is d related to d'?
how is v related to gamma?

d = γ(d' +Vt')

and v is related to gamma through the lorentz factor isn't it?

Simon Bridge
Homework Helper
Where did that V come from? Electric potential?

I think you need to revise length contraction - make sure you know what the equations mean... but yep ... ##(v/c)=\sqrt{1-1/\gamma^2}##

Careful about which is the primed and unprimed coordinates ... if d is the proper length, what is d'?

The time in the ship is T', the ship sees objects in the Earth frame passing at speed v=d'/T'... if the ship speed is constant.

HallsofIvy
Homework Helper

## Homework Statement

ESTIMATE the Lorentz factor of a spaceship which travels to a nearby star 10pc from the solar system in 50 years spaceship time.

## Homework Equations

Lorentz factor: γ = 1/√(1 - v^2/c^2 )

Time dilation: Δt' = γΔt

## The Attempt at a Solution

Using the distance and time given, from the spaceship's point of view, it is travelling at a velocity of 0.65c.
"From the spaceship's point of view", it isn't moving at all! The star is moving toward it at 0.65c.

I think I need to figure out the velocity from the point of view of Earth and then put that into the lorentz factor, however I don't know how to do that and that doesn't seem like an estimate to me.

I also thought that perhaps if I could work out the time it took to get there from the point of view of Earth I could rearrange the time dilation equation and get gamma from that.

I think my problem is I don't know what I can reasonably assume/estimate without getting this completly wrong.

Help :(

V is velocity of some sort, I think, not electrical potential.

I've just read up on length contraction. Proper length is the ship's length when it's at rest and in the reference frame where the spaceship is moving it will experience length contraction - Does this mean from the spaceship's point of view it travels less than 10pc?

d' = (1/γ)d - Where d is the proper length and d' is the length seen by the ship.

So d is 10pc because that is it's length at rest from the Earth's point of view.

What I still don't understand is how I can work out gamma only knowing the proper length and T'.

Simon Bridge
Homework Helper
V is velocity of some sort, I think, not electrical potential.

I've just read up on length contraction. Proper length is the ship's length when it's at rest and in the reference frame where the spaceship is moving it will experience length contraction - Does this mean from the spaceship's point of view it travels less than 10pc?
That is correct! Well done.

That's how the spaceship crew explain the fact that they got to the star in such a short time without the star having to go faster than light.

d' = (1/γ)d - Where d is the proper length and d' is the length seen by the ship.

So d is 10pc because that is it's length at rest from the Earth's point of view.

What I still don't understand is how I can work out gamma only knowing the proper length and T'.
But you have three equations and three unknowns. Hint: simultanious equations.

This way - the only assumption is that the ship/earth relative speed is constant over the journey.
Note: if the speed v is verry close to c, then d' is very close to how many light-years?
Then you can use the length-contraction formula to estimate gamma.

Last edited:
Okay I've tried playing around with the equations;
1) (v/c)=1−(1/γ2)
2) d=d'γ
3) t' = γt

The question wants we to estimate the lorentz factor so I need to use these equations to calculate velocity, correct?

I tried dividing 2) by 3) to get (d/t') = 10/50 = 0.2 parsecs per year but that is just using the Earth's distance and the spaceship time so it seems i've gone through all this stuff only to go back to the beginning :(

What am I doing wrong?

Simon Bridge
Homework Helper
The question wants we to estimate the lorentz factor so I need to use these equations to calculate velocity, correct?
You need to calculate gamma.

Spelling it out (and using a 0 subscript to indicate the frame in which the star is at rest.)
(1) ##(v/c)^2=1-1/\gamma^2##
(2) ##d_0=\gamma d##
(3) ##v=d/T=d_0/T_0## <---<<< you forgot...

>>>---> these simple relations still apply inside each reference frame; SR just relates frames to each other.

Note (1):
you can use the time dilation formula for #2 instead if you prefer.

Note (2):
vT is the distance to the star (in the ship-frame), cT is the distance light travels in the same time (same frame). If the relative speed v is very close to the speed of light, then these distances will be approximately the same. You could use this observation to get a back-of-envelope estimate for gamma using the length contraction formula (or the time-dilation formula) by itself.