The fastest route between two points

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• ddddd28
In summary: But after reading the OP, I don't think that's the case here. I think it's more like a "rate of travel" problem, where there are different speeds in different parts of the route.In summary, the problem involves finding the fastest route between two points, taking into account different velocities in different regions and a border between the points. The solution requires minimizing the overall duration of the route, which can be expressed as a function of a variable m. However, solving the equation for m leads to a fourth-degree equation, making it a challenging problem. This type of problem is commonly found in calculus textbooks, with variations such as refraction and light passing through different media.
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...

ddddd28 said:
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!

ddddd28 said:
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.

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.

pinball1970, SSequence, fresh_42 and 1 other person
I get : m = v1 √(d1^2+m^2) / ( v2 √(d2^2+(S-m)^2) + v1 √(d1^2 + m^2) )

Mark44 said:
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).

1. What is the fastest route between two points?

The fastest route between two points is the shortest distance between the two points, which is a straight line.

2. How do you calculate the fastest route between two points?

The fastest route between two points can be calculated using the Pythagorean theorem, which states that the shortest distance between two points is the square root of the sum of the squares of the differences between the coordinates of the two points.

3. Are there any factors that can affect the fastest route between two points?

Yes, there are several factors that can affect the fastest route between two points, such as traffic, road conditions, and terrain. These factors can cause the shortest distance between two points to change, resulting in a longer or slower route.

4. Can the fastest route between two points be different for different modes of transportation?

Yes, the fastest route between two points can vary depending on the mode of transportation. For example, a car may take a different route than a bicycle or a pedestrian due to one-way streets or designated bike lanes.

5. Is the fastest route between two points always the most efficient?

Not necessarily. While the fastest route may be the shortest distance between two points, it may not always be the most efficient. Factors such as tolls, traffic, and road closures can make a longer route more efficient in terms of time and cost.

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