Particle Motion (Astrophysics)

[Gregorski]

Homework Statement

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.

Homework Equations

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

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 = e^v₀ht+c
r = e^ce^v₀ht
e^c=R
r=Re^v₀ht
Plugging into (1)
Re^v₀ht = v₁t

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

 

[Andrew Mason]

Homework Statement

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.

Homework Equations

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

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 = e^v₀ht+c
r = e^ce^v₀ht
e^c=R
r=Re^v₀ht
Plugging into (1)
Re^v₀ht = v₁t

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

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

 

[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.

Greg