# Homework Help: (Re-Answered) Spacetime interval homework question?

1. Apr 22, 2012

### CCCP

1. State the problem, all variables and given/known data
The star Alpha goes supernova. Ten years later and 100 ly away, as measured by astronomers in the galaxy, star Beta explodes.

An alien spacecraft passing through the galaxy finds that the distance between the two explosions is 120 ly. According to the aliens, what is the time between the explosions?

s=Spacetime interval
c=Speed of light≈3 x 10^8 m/s or 1 lightyear/year
Δt=Change in time from reference of astronomers in the galaxy=10 years
Δx=Distance in light years between Alpha and Beta's explosions from reference of astronomers in the galaxy= 100 light years
Δx'=Observed distance between explosions observed by Alien spacecraft

Δt'=Observed time between explosions from the reference of the Alien spacecraft=?

2. Relevant equations
s^2=c^2(Δt)^2 - (Δx)^2 =c^2(Δt')^2 - (Δx')^2

3. The attempt at a solution

Plug in known variables to equaiton (Unit abbreviations- ly=light year; y=year)
=> (1ly/y)^2 *(10y)^2 - (100 ly)^2 =(1ly/y)^2*(Δt')^2 - (120 ly)^2

Because the spacetime interval s between the two events is invariant, we can use algebra you can get rid of the c term as it appears on both sides. The equation is simplified to:

=> (10y)^2 - (100y)^2 = (Δt')^2 - (120 y)^2

=>-9900y^2 = (Δt')^2 - 14400 y^2

=>(Δt')^2=4500

=>Δt'=67.08 years