Welcome to the Physics page for The Quarters.

Imagine yourself at the top of a playground climbing frame, you want to reach the bottom as quickly as you can. There is a choice of three slides with different tunnels: a straight tunnel, a curved tunnel and a spiral tunnel. Which do you currently believe is the fastest way to reach the ground? Now let me show you a discovery that might just change your mind.

In 1638, Galileo Galilei was the first mathematician to question this enigma. According to his calculations, the route that would create the fastest descent was an arc of a circle, but many questioned if this path really provided the least travel time. 

Soon later in 1696, the Swiss mathematician Johann Bernoulli posed a challenge to his other colleagues. The goal was to find the exact curve that would allow an object, sliding under gravity, to reach the end in the shortest time.

At first, many might think that a straight line is the optimum answer as it has the shortest distance. However, this is not true in terms of time. A straight ramp restricts the object, it is unable to increase its speed at a fast enough rate so it can reach the bottom the quickest. How do you find a route that ultimately provides the least journey time for a sliding object? 

When you have a steeper start, due to gravity, it allows an object to gain speed, this in turn helps reduce the total journey time of the object. However, if the path is too steep for too long, the extra distance will slow down the object. The challenge therefore was to find a route with the perfect balance between steepness and length so that when an object slides down the ramp it should reach the bottom the quickest. 

Cleverly, Bernoulli realised that this problem was similar to the nature of light with how it bends when it moves between different mediums; moving from air to water. If you think about light in a single medium, it will always travel in the shortest distance, but this is not the case when dealing with more than one medium with different densities. When light travels from one medium to another it bends and refracts in a way that doesn’t follow the natural law of light, it should always travel the shortest distance. This raised a lot of confusion, but later in 1657, a passionate mathematician named Pierre de Fermat tried to understand the nature of light, and the reasons behind why light was bending in ways that did not follow its own rules. Soon he realised that light does not follow the shortest distance, but in reality, it follows a path that takes the least amount of time. 

Interestingly, when Fermat proved his discovery, there was an evident connection to Snell’s Law. Fermat’s principle of least time, which explains that light always takes a specific path to minimise travel time, can additionally be illustrated using Snell’s Law. Snell’s Law states that the ratio of the angle of incidence to the angle of refraction (the angles at which light moves when travelling between different mediums) is equal to the ratio of the speeds at which light travels depending on the medium.

Bernoulli applied these two optical ideas to his problem but instead of light moving through different materials, he thought of an object moving through layers where the density of each layer decreased progressively. By picturing the situation like this it would mean that the angle of refraction would be greater than the angle of incidence for each layer, and hence it would create a shape somewhat similar to a curve. Just as light bends to minimise travel time, the falling object must now follow a curved path to minimise descent time. 

This curve is known as a cycloid curve or famously called the Brachistochrone curve which translates from Greek ‘shortest time curve’. From this, Bernoulli confirmed that the path which follows the fastest descent was a cycloid curve rather than a straight line or circular path. 

Think of a cycloid curve like this: if you mark a dot on the edge of a circular wheel and let it roll forwards, you will ultimately find that the path that the point creates is a looping arc rather than a straight line. This looping arc is known as a cycloid curve. 

Bernoulli’s discovery led to important discoveries in physics and mathematics, for instance a new branch of mathematics that finds the most efficient paths, later used in engineering and physics. Furthermore, The Principle of Least Action was introduced, which explains why nature often follows the most efficient path, influencing everything from classical mechanics to quantum physics.

To sum up, the problem of fastest descent is a great example of how different areas of physics are connected. Bernoulli used the study of optics to solve a problem about falling objects, which just shows the significant connection of nature and mathematics. Using this idea, it was soon discovered that the cycloid, which is traced by a rolling wheel, turned out to be the perfect solution. This discovery still influences modern physics, proving that whether it’s light bending, planets orbiting, or objects falling, nature always follows the best possible path.

References:

Rojo, A. and Bloch, A., 2018. The Principle of Least Action: History and Physics. [online] Available at: https://ve42.co/Bloch2018 [Accessed 9 May 2025].

Coopersmith, J., 2017. The Lazy Universe: An Introduction to the Principle of Least Action. Oxford: Oxford University Press. [online] Available at: https://ve42.co/LazyU [Accessed 9 May 2025].

3Blue1Brown, 2020. The Brachistochrone, with Steven Strogatz. [video] YouTube. Available at: https://ve42.co/Brach3B1B [Accessed 9 May 2025].

Vsauce, 2017. The Brachistochrone. [video] YouTube. Available at: https://ve42.co/BrachV [Accessed 9 May 2025].