# Space Expedition

## Situation

The International Space Agency is planning to send a colonization expedition to an exoplanet 10 light years away. The expedition will be split into two ships (a single ship would be too large to be feasible), and due to funding issues the launches of the two ships will have to spaced out by 40 years, the first ship (Spaceship A) launching in 2135, and the second ship (Spaceship B) launching in 2175. Each of the ships also carries a space probe that will separate from the ship upon reaching the exoplanet and continue the voyage further past into the distance. Current projections (2015) expect the speed of spacecraft to double every 40 years, from the current speed (2.5% the speed of light). Which spaceship should hold the first settlers on the exoplanet, and which spaceship should hold the backup supplies?

## Variables and Equations

Defining Variables

let d represent the distance of the space probe (the spaceships stop after 10 light years) from Earth, in light years

let t represent the time, in Common Era years

light travels 1 light year per year, thus light would have a slope of 1

a percentage of the speed of light (e.g. 50%) would be the same as the slope (50%, or 1/2)

the x-intercepts (or t-intercepts) are the launch dates of the spaceships, and can be used in conjunction with the lines' slopes to find their y-intercepts (or d-intercepts)

d will be represented on the y-axis

t will be represented on the x-axis

Note: As there cannot be negative values of d, in the equations "d" will be referred to as (√d)², which is mathematically equal to d, however if the value of d is negative, √d becomes an imaginary number and thus the line terminates for all values of d below zero.

Equations

Spaceship A

(√d)² = 1/10 t - 427/2

Spaceship B

(√d)² = 1/5 t - 435

## Point of Intersection

Before the point of intersection at (2215, 8), the purple line (spaceship probe A) is further away from Earth than the blue line (spaceship probe B).

When t < 2215, probe A's distance > probe B's distance

Before the point of intersection, probe A reaches a certain distance before probe B reaches the same distance

When d < 8, probe A's time < probe B's time

After the point of intersection, the blue line (spaceship probe B) is further away from Earth than purple line (spaceship probe A).

When t > 2215, probe A's distance < probe B's distance

After the point of intersection, probe A reaches a certain distance after probe B reaches the same distance

When d > 8, probe A's time > probe B's time

At the point of intersection, probe B is equally far from the Earth as probe A, and as probe B has a greater slope (is travelling faster), at the point of intersection probe B is in the process of passing probe A.

Whichever spaceship reaches a distance of 10 light years first will reach the exoplanet first, and will carry the first settlers. Remember this statement:

When d > 8, probe A's time > probe B's time

With a distance of 10 in mind, d is greater than 8, thus probe A will reach 10 light years after probe B, so probe B will be the first to reach the exoplanet.

## Summary

In the year 2215, as both space probes are 8 light years away from Earth, probe B passes probe A, and reaches 10 light years (the exoplanet) first, despite being launched later.

Therefore, spaceship B would have to carry the first settlers to colonize the planet, and spaceship A would deliver the second round of supplies.