# What Mathematics questions can be better solved with concepts from Physics?

Over the years, I've seen several questions in mathematics that can be solved using concepts borrowed from Physics. Having seen these question, I'm interested to find out what other mathematics questions you've found that can be better solved with a concept from physics - or at least where the application of physics is interesting and perhaps illuminating.

Examples
One of these questions is on minimizing the time taken for a lifeguard to go out to a stationary distressed swimmer. In the scenario, the lifeguard runs faster than he swims in the water, and as such the straight line is not the fastest way for the lifeguard to reach the swimmer. Students will normally use calculus to solve this problem, and the answer can be obtained after some work - however, a much more convenient (and intuitive) way is to borrow from the idea of refractive indices in geometric optics. We recast the situation by replacing the beach and the sea with two materials with different refractive indices, choosing the appropriate refractive indices proportionate to the ratio between the lifeguard's velocities while running and swimming. The problem is then reduced to finding a beam of light that passes through both the swimmer and the lifeguard's position. (For a more complete explanation, you can visit this site: http://findingmoonshine.blogspot.sg/2012_05_01_archive.html)

Another of these questions requires one to prove that, in an acute-angled triangle, the angle subtended by any side of the triangle at the Toricelli point is 120°. Again, instead of using trigonometry, one can use the concept of hanging equal weights from a (frictionless) string at each of the vertices of the triangle, and then tying each of the three strings together at one knot placed on the surface of the triangle. The equilibrium position of the knot is the Toricelli point, and one can then complete the proof by considering forces acting on the knot.

Looking forward to hearing from you!

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 This post asking for the connections between math and physics might also be of interest. – amWhy Dec 10 '12 at 14:39 Just noting, your example is pretty "circular" because to prove the property of refractive indices you do the calculus you mention. So its not really a valid example unfortunately. – dinoboy Dec 10 '12 at 17:46 @dinoboy are you referring to the principle of least time? – Vincent Tjeng Dec 11 '12 at 11:28 No, the principle of least time requires no time. However, the route derived using the principle of least time requires calculus to prove it is indeed the route of least time I believe. – dinoboy Dec 11 '12 at 17:33

Volume of Sphere deduced by Archimedes using mechanics concepts: Archimedes methold.

In my opinion the Archimedes method to deduce the volume of the sphere is a beatiful examples of applications of physics concepts to a mathematical proof. I will not plagiarize the method describing it here but I will indicate sites that describe the method.

I hope helped.

See the Archimedes method in this site, Youtube and pr$\infty$f wiki.

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Minimal surfaces using, for example, soap bubbles.

Maximum flow using actual pipes.

Optimal space filling and tesselations inspired by honeycombs.

Solving some of equation sets can be done via circuits of resistors.

There were analog computers build to analyze differential equations.

Probability can be calculated by experiments, dice games were examples of this.

Moreover, a lot of regular operations like addition, multiplication, squares and roots, powers, integration, etc., can be done by experiments.

Finally, modern computers use physics (i.e. are real) and can solve a variety of math problems :-)

Cheers!

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