 # Problem of the Week Problem B and Solution Are We There Yet?

## Problem

Planets travel around the sun in elliptical orbits (ovals). Mercury is the planet closest to the sun. The distance between Earth and Mercury ranges from $$77\,000\,000\,$$km at its closest distance to $$222\,000\,000\,$$km at its farthest distance.

Because distances are so great in the solar system, scientists measure them in Astronomical Units, or AU. One AU is equal to the average distance between the Earth and the Sun, or about $$149\,600\,000\,$$km.

Complete the missing information in the table below.

Planet Distance in AU from Earth Distance in km from Earth Travel Time
Mars $$0.52$$
Venus $$61$$ days
Saturn $$1\,275\,000\,000$$
Neptune $$29.09$$

To calculate the travel time, assume you are travelling from Earth to the planet in a rocket at a speed of $$28\,000\,$$km per hour throughout your flight. Pick the most reasonable unit of measure for time (for example, $$15\,000$$ hours doesn’t mean much, but when divided by $$24$$ to get $$625$$ days, you know that it’s almost $$2$$ years).

## Solution

The completed table is shown below.

Planet Distance in AU from Earth Distance in km from Earth Travel Time
Mars $$0.52$$ $$77\,792\,000$$ $$2\,778\,\textrm{hr}=116$$ days
Venus $$0.27$$ $$40\,992\,000$$ $$61$$ days
Saturn $$8.52$$ $$1\,275\,000\,000$$ $$45\,536\,\textrm{hr}=1897\,\textrm{days}=5+$$ years
Neptune $$29.09$$ $$4\,351\,864\,000$$ $$155\,424\,\textrm{hr}=6476\,\textrm{days}=17+$$ years

Since $$1\,\textrm{AU}=149\,600\,000\,\textrm{km}$$, to convert from the distance in AU to the distance in km (for Mars and Neptune), we multiply the distance in AU by $$149\,600\,000$$. Similarly, to convert from the distance in km to the distance in AU (for Saturn), we divide the distance in km by $$149\,600\,000$$. This allows us to fill in both distance columns for Mars, Saturn, and Neptune.

To calculate the travel time, we use the speed of the rocket, which is $$28\,000\,$$km per hour. If we divide the distance in km by $$28\,000\,$$km per hour, we will get the number of hours it takes to travel that distance, which is the travel time. We can then convert this to a more appropriate unit as we see fit.

To calculate the distance in km from the travel time (for Venus), note that $$61$$ days is equal to $$61\times 24= 1464$$ hours. Thus, travelling at $$28\,000\,$$km per hour, the distance covered would be $$28\,000\times 1464=40\,992\,000\,\textrm{km}$$. We can then convert the distance in km to the distance in AU as we did for Saturn.