Relativity: Time Dilation and Length Contraction

[SeannyBoi71]

Homework Statement

Two events are observed by inertial observer Stampy to occur a spatial distance of 15 c·s apart with the spatial coordinate of the second larger than the spatial coordinate of the first. Stampy also determines that the second event occurred 17 s after the first. According to inertial observer Philip moving along Stampy’s +x axis at unknown velocity v, the second event occurs 10 s after the first. (1 c·s = 1 light-second = unit of distance.)a) Given Philip measures the spatial coordinate of the second event to be larger than the first, determine v.
b) How far apart spatially (in c·s) do the two events occur according to Philip?
c) Does there exist an inertial reference frame v < c in which the second event can occur before the first? Briefly explain in one sentence at most.
d) Inertial observer Kenny observes the proper time between the two events. How fast along Stampy’s +x axis does Kenny move?

Homework Equations

γ=1 / √(1- v²/c²)
Δt=γΔt₀
l=l₀/γ
(Δs)² = c²(Δt)² - (Δx)²

The Attempt at a Solution

I have so far worked out all parts but d) and need to check these answers.
a) I took Δt=17 and Δt₀=10 so that dividing I could get γ=1.7. Was I right in making the assumption about which time interval was which? The only reasoning I have as to why I did that is because dividing the other way around gives me a number <1. But next I plugged in 1.7 to the Lorentz factor equation and solved for v=0.81c.
b)The formula is called length contraction, and since Phillip is moving relative to Stampy, distances are contracted for him (i.e. are shorter) so I did l=l₀/γ = 15cs/1.7=8.8 cs.
c)I calculated that the spacetime invariant > 0 and so the two events are time-like separated, and all observers must agree on the time-ordering.
d) This part confused me. I made the assumption in a) that the proper time was 10 seconds, which is measured by Phillip, which leads me to think I was wrong making that assumption.

 

[BruceW]

I don't think you've done it right. In stampy's reference frame, the two events happen in different places. So you can't use the equation Δt=γΔt₀. You need to use the more general Lorentz transform for when the events happen in different places and different times.