🦋 Shortest time to reach a destination. Science or mathematics?

What is the direction we have to travel if we want to go from a point A to a point B in the shortest possible time?

Is the above an easy question that has a simple answer : a straight line AB?

If we travel at a constant speed the answer is yes. If our speed depends on the path we take, the answer is no.

The answer require complicated mathematics  and is related to science.

Suppose that A is x meters from the age of a pool. B is a point in the pool that is y meters away from the edge of the pool. The line AB is not perpendicular  to the edge of the pool.

You can walk at the speed of u and swim at the speed of v 

What is the route you have to take to reach B in the shortest possible time?

If u = v, the answer is a straight line AB.

If u is not equal to v you will have  to walk to a point C before swimming to B

How do you find C and what angle do AC  and CB  make with the edge of the pool?

The answers  depend on the ratio between u and v.

The time taken to go from A to C  is AC/u and the time taken to go from C to B is CB/v.

Your immediate  problem is that you do not know how long AC and CB are.

So you have to draw a perpendicular lines AD and BE to the edge of the pool.

AD = x and BE = y

Let us call the length of the line DE as f and DC as g

CE will then be = f - g

We now have 2 right angle triangles ADC and BCE.

We also have the lengths of 2 sides of each of these 2 triangles.

Using Pythagorous fomula, we can calculate AC and CB. 

Next we are ready to calculate the time taken (T) to reach B

T = AC/u  + CB/v

So far we have only used basic mathematics that secondary school students are capable of.

To find the minimum  value of T and where C is, we will have to use more advanced  mathematics including  calculus and algebra to find the roots of a  quartile or 4th power equation.

Calculus allows us to find maximum and minimum  values by using a technique  known as "Differentiation"

By looking at the rate of change of T we can find the minimum value of T. This is when the rate of change of T changes from negative to positive. This is when the rate of change  of T is  zero.

Most of the readers here is probably not familiar  with calculus so I will explain in layman's  terms how it solve our problem.

The position of C will determine the total time taken.

As we change the position of C and increase the length of AC (starting from zero), the rate of change of T is negative.

So the time taken to reach B will keep falling.

After a while the rate of change of T becomes positive and the T increase instead of decrease. 

The minimum value of T is at the turning point when the rate of change of T is zero.

What has the above to do with science?

The question above is like looking at the path light takes when changing from 1 medium to another. Light will change direction at the border of the 2 mediums  eg air and water.

This change in direction is called REFRACTION. Snell's law allows us to calculate how much light rays will change direction. (The angle of incidence and the angle of refraction)

The amount of change of direction depends on the speed light travels in the 2 medium. Light travels faster in air than in water.

Different types of glass refracts light to different extent. Glass with a higher Refractive Index (RI) refracts light more.

People who wear spectacles to correct high degrees of short sightedness  or long  sightedness  will appreciate  high RI glass. The use of high RI glass make the lenses  a lot thinner and lighter. The spectacles do not look look so ugly and are more comfortable  to wear.

Now let us look at REFLECTION.  This is when the value of y in the above question is negative.

After reaching C, light does not move in the new medium but bounce back (is reflected) to the original medium.

INTERESTING FACT :

The path that light takes is the path that gives it the shortest possible time to reach its destination. 

Both the laws of REFRACTION and REFLECTION  is based on the path light should take to in order to achieve the shortest possible time

CONCLUSION:

God is an efficiency  expert 😃🤣🦋 and am I a mathematician or am I a scientist?