# Particle Motion (Astrophysics)

Tags:
1. Jan 31, 2017

### Gregorski

1. The problem statement, all variables and given/known data
This is new for me, so forgive me my clumsiness. I am working on the following problem:
A particle p is moving with a velocity v1 = c (speed of light) towards an object q, which is moving in the same direction with the speed v2, where v1>v2. Now, v2 is a function of the distance r between p and q. I need to find the time when p "catches up" with q.

2. Relevant equations
v2(r)= v0hr, where h - the Hubble constant, v0 - initial velocity, and r - distance between p and q

3. The attempt at a solution
Particle p will "catch up" with q when the distance between them is 0, so we have:
r - s =0 and r = s (1)
v2(r)= v0hr
Differential equation:
ds/dt = v0hr

Separating variables:
ds/r= v0h dt
Integrating:
log r = v0ht + c
r = ev0ht+c
r = ecev0ht
ec=R
r=Rev0ht

Plugging into (1)
Rev0ht = v1t

I am not sure how to proceed from here or if any of it makes sense.

2. Feb 4, 2017

### Andrew Mason

Welcome to PF Gregorski!

There are two reference frames. The observer frame, relative to which q is moving at speed v2, and q's rest frame. It is not clear from the question as to the reference frame in which r or time is being measured.

According to Special Relativity, p must have 0 rest mass and must be moving at speed c relative to all inertial reference frames. The observer and q would measure the time for p to reach q differently.

AM

3. Feb 4, 2017

### Gregorski

Andrew,
Thank you for your input; you're absolutely right there are two frames. I managed to do the last step by applying Lambert W function.
Best,
Greg