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But how?Stephen Tashi said:What level of mathematics are we allowed to use in solving this problem? Calculus?
It seems simplest to begin by finding distance DO.
You didn't answer this question. It's important.Stephen Tashi said:What level of mathematics are we allowed to use in solving this problem? Calculus?
I don't think your attempt to fit this problem into Snell's Law is correct. You need to write the equations for the time to the pavement as a function of angle, and the time to the final point on the pavement starting at that landing point. If you can use calculus, you just minimize the total time as a function of the angle you take to the pavement.En Joy said:But how?
En Joy said:But how?
Come to think of it, it might end up looking like Snell's Law. I'll have to try deriving it...berkeman said:I don't think your attempt to fit this problem into Snell's Law is correct
I know calculus and the derivation of Shell's law because I have completed graduation (Physics honours) and I am a primary school teacher also. The problem came to my mind suddenly while studying calculus of variation, so this is not a school homework. Trust me.berkeman said:You didn't answer this question. It's important.
I don't think your attempt to fit this problem into Snell's Law is correct. You need to write the equations for the time to the pavement as a function of angle, and the time to the final point on the pavement starting at that landing point. If you can use calculus, you just minimize the total time as a function of the angle you take to the pavement.
I think the analogy with Snell's Law is good. Did you look through the links? The minimum time condition seems very applicable to the problem you came up with...En Joy said:I know calculus and the derivation of Shell's law because I have completed graduation (Physics honours) and I am a primary school teacher also. The problem came to my mind suddenly while studying calculus of variation, so this is not a school homework. Trust me.
Yes, calculus will need here to sove this problem but I tried my best and did already show my attempt in the picture.
Yes, I have read the link. It illustrates nothing but the derivation of Snell's law using calculus.berkeman said:I think the analogy with Snell's Law is good. Did you look through the links? The minimum time condition seems very applicable to the problem you came up with...
But that's what the derivation of Snell's law is about -- the path with the least time...En Joy said:find the path of least time to reach your friend?
I want to know the value of θ_{m}berkeman said:But that's what the derivation of Snell's law is about -- the path with the least time...
En Joy said:I know calculus and the derivation of Shell's law because I have completed graduation (Physics honours) and I am a primary school teacher also. The problem came to my mind suddenly while studying calculus of variation, so this is not a school homework. Trust me.
Yes, calculus will need here to sove this problem but I tried my best and did already show my attempt in the picture.
En Joy said:I want to know the value of θ_{m}
En Joy said:Yes, I have read the link. It illustrates nothing but the derivation of Snell's law using calculus.
I actually want to know something like this. Suppose you are in position A(in mud) and your friend is at B(in pavement) as in picture. Now how can you find the path of least time to reach your friend? All the distances and the two angles are known(see the main image).
I think may be Snell's law and some calculus may help in this case but I could not try further.
Please solve this completely.
THANK YOU VERY MUCH @Ray VicksonRay Vickson said:The solution is pretty horrible.
First, I will change the picture to put point A on the y-axis at ##(0,h)## and point B to the right at ##(a,-b)##. So, the path crosses the horizontal axis at ##x \in (0,a)##. (Basically, instead of putting the origin at the crossing point and the having variable endpoints, I prefer to fix the ends---but of course, it is equivalent.) We have two equations. Since ##x = h \tan(\theta_m)## and ##a-x = b \tan(\theta_p)## we have
$$h \tan(\theta_m) + b\tan(\theta_p) = a.$$
We also have ##\sin(\theta_p) = r \sin(\theta_m)##, where ##r = v_p/v_m##. Expressing everything in terms of ##z = \sin(\theta_m)## the problem reduces to the solution of the equation
$$ (1)\;\;\frac{h z}{\sqrt{1-z^2}}+\frac{b r z}{\sqrt{1-r^2 z^2}} = a.$$
This can be re-written as a rather horrible 4th degree polynomial in ##u =z^2## that is too long and complicated to reproduce here. Using 4th degree solution formulas, the equation can be solved in principle, but the results are not pretty. Maple takes about 59 pages of nasty, and complicated formulas to write out all four solutions to the polynomial. I'm not sure what use that would be to anybody.
Note added in edit: a much easier equation results if we look at the crossing point ##x## as the variable, instead of the angles. Snell's law gives
$$(2) \;\;\frac{1}{v_m} \frac{x}{\sqrt{x^2+h^2}} = \frac{1}{v_p} \frac{a-x}{\sqrt{(a-x)^2+b^2}}.$$
Squaring both sides leads to a relatively tractable 4th degree polynomial. Exact solutions are still lengthy and un-enlightening. However, numerical solution of the ##x##-problem is simpler than for the ##\theta_m##-problem. Eq. (2) also follows directly by setting the ##x##-derivative to zero of the total time function ##T = \sqrt{x^2+h^2} / v_m + \sqrt{(a-x)^2 + b^2} / v_p.##
To determine the quickest path, you will need to consider the distance, time, and obstacles present in the path. You can use mathematical equations or algorithms to calculate the shortest route.
Some factors to consider when finding the quickest path include the mode of transportation, traffic conditions, distance, and any potential obstacles or detours along the way.
There are various methods and algorithms that can be used to find the quickest path, depending on the specific situation and needs. Some common ones include Dijkstra's algorithm, A* algorithm, and the Floyd-Warshall algorithm.
Yes, there are many technological tools and applications available that can help you find the quickest path. These include GPS devices, map apps, and route planning software.
You can determine if the path you have chosen is the quickest one by comparing it to other potential paths and considering factors such as distance, time, and obstacles. It may also be helpful to use technology or consult with experts in the field.