The fastest route between two points

[ddddd28]

Hello,
Consider the next scenario:
I wish to travel from point a to point b as fast as possible. Between the points there is a border. In the region from a to the border, I can move only with velocity v1, and after the border, I am allowed to move only with velocity v2. All the additional details are given in the sketch:

As it turns out, the problem is not as easy as it appears to be because it is not clear where I should pass the border. If v1 is bigger than v2 then I should consider taking a longer route in the first section to compensate on the time spent in the second section. However, a longer route in the first section adds up to the overall distance.
My attempt to solve it was to express the overall duration of the route as a function of m and minimize it. To my dismay, I ended up with a fourth-degree equation that didn't want to crack...

 

[pinball1970]

Hello,
Consider the next scenario:
I wish to travel from point a to point b as fast as possible. Between the points there is a border. In the region from a to the border, I can move only with velocity v1, and after the border, I am allowed to move only with velocity v2. All the additional details are given in the sketch:
View attachment 247182
As it turns out, the problem is not as easy as it appears to be because it is not clear where I should pass the border. If v1 is bigger than v2 then I should consider taking a longer route in the first section to compensate on the time spent in the second section. However, a longer route in the first section adds up to the overall distance.
My attempt to solve it was to express the overall duration of the route as a function of m and minimize it. To my dismay, I ended up with a fourth-degree equation that didn't want to crack...

No replies as yet so I will ask a question.
There will be different answers on this depending on which v is faster and by how much. Have you given all the details on this?
Some trig scenarios in there to play with.
I'm not a maths guy but I like the subject. @fresh_42 and @Mark44 will probably have 2 or 3 steps to solve!

 

[phinds]

If v1 is bigger than v2 then I should consider taking a longer route in the first section to compensate on the time spent in the second section.

I don't get at all why that should be an issue. The solution needs to be general and has to encompass both situations automatically. I agree that it's a nasty problem but not because you can't decide in advance which path should be longer. That's what you are supposed to be SOLVING for, really.

 

[Mark44]

This is a standard problem in calculus textbooks, where the goal is to minimize the time rather than the distance covered. Typical problems include scenarios in which you need to get to a point on the other side of a river, and your velocity while swimming or boating is different from your velocity on foot, as well as finding the path of least time for light passing through two different media.

 

[Aian Khan]

I get : m = v1 √(d1^2+m^2) / ( v2 √(d2^2+(S-m)^2) + v1 √(d1^2 + m^2) )

 

[SSequence]

Typical problems include scenarios in which you need to get to a point on the other side of a river, and your velocity while swimming or boating is different from your velocity on foot, as well as finding the path of least time for light passing through two different media.

Yes, refraction and the principle of least time was the first thing that came to my mind (since the kind of diagram, as in OP, is often associated with it).