- #1

Gregorski

- 8

- 0

## 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

*v*(speed of light) towards an object

_{1}= c*q,*which is moving in the same direction with the speed

*v*, where

_{2}*Now,*

*v*_{1}>v_{2}.*v*is a function of the distance

_{2}*r*between

*p*and

*q.*I need to find the time when

*p*"catches up" with

*q*.

## Homework Equations

*v*where

_{2}(r)= v_{0}hr,*h*- the Hubble constant,

*v*initial velocity, and

_{0}-*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)

*v*_{2}(r)= v_{0}hr*Differential equation:*

ds/dt =ds/dt =

*v*_{0}hrSeparating variables:

*ds/r=*

*v*_{0}h dtIntegrating:

*log r =*

r = e

r = e

e

r=R

*v*_{0}ht + cr = e

^{v0ht+c}r = e

^{c}e^{v0ht}e

^{c}=Rr=R

*e*^{v0ht}Plugging into (1)

*R*

*e*^{v0ht}= v_{1}tI am not sure how to proceed from here or if any of it makes sense.