Blog #26, Worksheet #8.1, Problem #1, Hubble Flow Thought Experiment

1. Before we dive into the Hubble Flow, let’s do a thought experiment. Pretend that there is an infinitely long series of balls sitting in a row. Imagine that during a time interval \(\Delta t\) the space between each ball increases by \(\Delta x\) 

(a) Look at the shaded ball, Ball C, in the figure above. Imagine that Ball C is sitting still (so we are in the reference frame of Ball C). What is the distance to Ball D after time ∆t? What about Ball B?
Let's call the original distance between Ball C and Ball D, x. Then we know that after a time \(\Delta t\) the ball travels a distance \(\Delta X\). Thus our total distance after a time \(\Delta t\) is \[d = x + \Delta x\]Because every ball is evenly spaced, we can assume that Ball B travels the same distance as Ball D.
(b) What are the distances from Ball C to Ball A and Ball E? 
Both Ball A and Ball E begin at a distance \(2x\) away from Ball C and over the course of time \(\Delta t\), the each ball has traveled a distance \(\Delta x\) away from the ball next to them which has also traveled a distance \(\Delta x\) from Ball C. Giving the total distance of Ball A/E away from Ball C as \[d = 2x + 2 \Delta x\]
(c) Write a general expression for the distance to a ball N balls away from Ball C after time ∆t. Interpret your finding. 
Seeing the pattern from part a) and b) we can easily generalize our equation for a ball N balls away as \[d = Nx + N \delta x\]
(d) Write the velocity of a ball N balls away from Ball C during ∆t. Interpret your finding.
Velocity is simply distance over time and we know that the distance a Ball N balls away from travels is \[d = N \Delta x \]Thus our velocity comes out to be \[v = \frac{N \Delta x}{\Delta t} = Nv\]