## Tuesday, January 10, 2017

### How Long Would It Take To Travel Around Earth Going 4 Miles Per Second?

Have you ever wondered how much time you would have to set aside to travel around the Earth while traveling 4 miles per second?

Is this a normal question that everyone asks themselves?

In the paragraphs below, I will entertain the first question while using dimensional analysis to arrive at an answer.  As for the second question, I will shed some light on the matter.

# Missile Technology

Over the last few years, North Korea has been testing various stages of nuclear weapons.  The end game would be (supposedly) into a rocket that would fly over the ocean and into the backyard of America.

Is North Korea close to achieving its goal?

You would have to ask the professionals in the field of 'foreign policy' and nuclear analysts about this topic.  At this point, you have probably guessed that the reason I ask the questions above is to get a grasp on the aerial capability and reach of various missile defense systems.  In the current post, I will restrict myself to the 'ground based missile (GMD) systems designed to thwart off an attack by countries like North Korea and Iran.

The question in the title arose out of an article in the 'Los Angeles Times' newspaper titled "The nation’s missile-defense system has serious flaws. So why is the Pentagon moving to expand it?" in which the author discussed the need for tremendous investment into long range missile technology at a very steep cost.  Here is the paragraph that caught my eye:

The interceptors are three-stage rockets, each with a 5-foot-long “kill vehicle” at its tip. In the event of an attack, interceptors would rise from their underground silos and soar toward the upper atmosphere. The kill vehicle is designed to separate from its rocket in space, fly independently at 4 miles per second and crash into an enemy warhead.

Over the course of decades, the US has designed missiles with precision and accuracy that is unparalleled.  Currently, lawmakers are concerned with the cost of the program which is weighed against the early results of the GMD system tests.  The cost of the GMD program (as it is called) is estimated to be \$4 billion.

To date, no decision has been made on exactly where the system will be installed.  Regardless, the geographic region in which the system is finally constructed stands to receive a boost to the local economy in the form of jobs and money to the region.   Therefore, the bidding process is a huge deal by politicians on the local and federal level.

The threat level is credible with the recent (as in today's news) article on the website 'CNN' titled "North Korea sends message to Trump amid threat to fire missile 'at any time'" with the following introduction:

North Korea says it could launch an intercontinental ballistic missile "at any time," even as Pyongyang appeared to offer Donald Trump an avenue for future talks.

Tensions on the Korean peninsula have risen considerably since leader Kim Jong Un said in his new year's message that the country was close to testing an intercontinental ballistic missile (ICBM) capable of delivering a nuclear weapon to the US mainland.
In a statement Sunday, a spokesman from North Korea's foreign minister said "the US is wholly to blame" for the development of its missile program.

According to the news, certain countries like North Korea are moving forward with their missile technology and have the U.S. in their sight as a target.  With the advent of spaceflight over the past few decades by various countries, the following question arises with regard to the development of missiles traveling around Earth:

Why is designing a missile so difficult when rocket technology is developed to send cargo into space?

The answer has primarily to do with the accuracy and target.  Distance is an issue.  First, the missile has to have enough power to reach the target.  Second, the accuracy is critical toward landing in a geographic region.   At first sight, these considerations might seem trivial.  Although, in the next section, designing a missile to counter another missile traveling at speeds of 'miles/seconds' is shown to be quite difficult.

Imagine the on board sensor system alone which must react to the oncoming missile -- in our case to destroy a threat.  The system must react toward any changing flight of the oncoming missile.  This might seem trivial if the target was 'static' -- i.e., not moving.  But at speeds of 4 miles per second, the accuracy is very difficult to achieve.

## A Single Lap Around Earth?

After reading this article, I could not get the speed at which the missile will fly out of my head.  In fact, my mind started to wander in search of a "metric" to use to analyze the speed with.   The object that finally resonated with me was the Earth.  At the speed of 4 miles per second, distances have to be fairly large to compare the value to.  Otherwise, the answer might not make very much sense.

With that being said, I decided to go through the dimensional analysis.  Below, I will walk you (the reader) through the calculations necessary to arrive at an answer.  To start with, the circumference of the Earth is needed.  That is the distance to travel around the Earth in a single trip.  If we consult the 'wikipedia' page for Earth, the value of the circumference is stated as: 24,901.461 miles at the equator (the middle).

Next, in order to determine the time that is required to travel that distance, we need an expression for velocity.  Shown below is a simple expression for linear velocity:

Note: in the current calculation, I calculated the time based on an assumption of a linear system -- neglecting gravitational forces along with other considerations.

With the expression for velocity above, there is "delta x" in the numerator and "delta t" in the denominator.  The initial question that was asked was the time that would be required to travel a given distance (around the Earth) at a certain velocity.  Therefore, to solve for time, the above expression for velocity needs to rearranged to solve for time as shown below:

Now, we have an expression that will give the time required if we input the distance and velocity.

To solve the expression above, the distance (circumference of the Earth) is inserted along with the velocity of the traveling object as shown below:

Wow! The result states that if a rocket is traveling around the Earth at 4 miles/second, the ship will take 6,225 seconds to travel around the Earth one time.  For those of us who do not think in 'seconds', how do we reconcile 6,225 seconds?  A unit conversion is required from units of 'seconds' to units of 'hours' as shown below:

The result above makes more sense to the average person's perception of time.  To travel around the Earth in a rocket at a velocity of 4 miles/second, a single trip will take 1.7 hours -- just under 2 hours.

What does the result suggest about the content of the article above -- missile defense?

Can you imagine trying to design a missile to travel at a speed of 4 miles/second?

Further, can you imagine trying to imagine designing that missile to hit a moving 'counter missile' from an enemy?

The logistics and precision behind these missiles is absolutely amazing.  The cost and difficulty matches the challenge.  Having a threat from a country with nuclear missiles is a serious issue if you are on the list of targets.  Therefore, investing in technology is crucial toward countering such a missile that is aimed at our country.

North Korea has had success in testing nuclear weapons last year.  I wrote a blog last year describing the force of such a weapon.  Now, imagine the force contained in a flying missile coming at our country?  Quite difficult, but scary at the same time.  Hopefully, the current technology is available to counter such a threat.

### Conclusion

Just imagine if your car or bicycle could travel at 4 miles per second.  You could circle the Earth in just under two hours.  Now that you have the method by which to complete the calculation above, you can choose other distances:

How long would a missile take to reach the U.S. from India?

How long would a missile take to reach the U.S. from North Korea?

You can verify the answers that I calculated:

1) India to United States = 8,431 miles -- therefore, a missile would reach the U.S. in 35 minutes.

2) North Korea to United States = 6,423 miles -- therefore, a missile would reach the U.S. in 27 minutes.

Wow! That is not a lot of time to defend oneself from a nuclear threat.  Precision and dependability are of the up most importance.

In the article above, the estimation of cost to update existing missile system is pegged at \$40 billion. Regardless of the avenue ahead, the technology (missiles) that is being developed is now better understood by carrying out the above calculation.  Having a grasp on the difficulties and cost of current technology allows us as citizens of the United States to influence policy making in a more informed manner.