Moon Rover Problem (An engineer estimation Problem)

本小篇幅报告为全英文，因为其第一读者为英文母语者。

How much time does it take for a moon rover to take pictures with 3m*3m size for the whole surface of the moon? The pictures are limited to the position near to the rover (3m*3m, or 1.5m for one side).

The assumption is that the moon rover goes along the latitude line and take pictures, and it starts from North pole and end at South pole. After finishing one round around the moon at a certain latitude, the rover turns 90 degree,goes along the longitude line for 3 meters to another track at a lower latitude, and goes around the moon along the latitude line again. The moon rover does this cycle all the way until it arrive at the South pole. Each track has a width of 3 meters because the pictures have a size of 3meter*3meter.

We found some basic information of the Moon (from wikipedia), they are shown below:

Radius: 1737 km  Circumference: 10914 km  Surface area: 37.9 trillion square kilometer

Speed of moon rover: 13km/h

First, we need to know how many rounds the rover has to go in order to cover all the surface area of the Moon. It is:

               Half the circumference/track width= (1737 km* )/3m

                                            =1737km**(10^3m/km)/3m

                                            =1818982 rounds

Second, we need to know how long each track is. Obviously, the length of one tracks is:

               L=2*R*sin  where R is the radius of the moon and  is the angle between the earth’s axis that pass through north and south pole and the radius that connect the center of moon and that track.

Last, we just need to sum each of the length of the track (which is L) up and find the total distance the rover needs to

Go, and here we use sigma sign (a series):

L*=  where L* stands for the total distance the rover’s going to cover,

                       R stands for the radius of moon,  has the same meaning as before.

We know the length of an arc S is :

                            S=r*   where  is the central angle, R is the radius of the circle.

At the track that we mentioned before, when the angle between the axis of moon and the

radius that connect the track and moon’s center is , so:

                The arc length from the North pole point to the track is:

                        S= R*   R is the radius of the moon

But S also equals to the sum of all the width of the tracks on that arc:

                         S= 3m*N           when N is the number of the track that we are looking at. m is just “meter”.

So, R*= N*3m, and:

                = 3*N/R= (3m*N/1737km)* (10^-3m/km)

So the series becomes this:   (we plugged  in)

It can be very hard to calculate this series, but we can try Maclaurin Expansion.

2  +  - )

Still the calculation complexity is way too large, but this is will be a very good estimation if the

value can ever be calculated.

By referencing to the estimation method which simply divide it by the speed of the rover and get

the time needed. Perhaps the answer is around 100,000 years.