BRACHISTOCHRONE

In this exhibit we have two points connected by two paths – one that is straight and the other a curved one. When we allow the masses to slide down simultaneously along these two paths, which of the two reaches the bottom end first?

Surprisingly, its not along the straight path! The curved path followed here is a cycloidal path. Newton showed that of all the paths connecting two points, motion along the cycloidal path would take the least time provided the motion is aided by gravity alone.

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What is a Cycloid? A cycloid is the curve described by the motion of a point on the edge of a circular wheel as the wheel rolls along a straight line.   

In this model we can see a straight path and a curved path. The surface of both the paths are equally smooth. There are two identical masses that can move along these paths. The two ends of both the paths are at different vertical heights from the ground level. Also, the difference in the vertical heights of the ends of one path is same as that of the other.

If the masses are simultaneously from the top ends of both the paths, which of the two would reach the bottom first ?

Many would think of the straight path. Therefore a straight line is the shortest distance between any two points in a plane. If the two points are joined by any curve, the distance between those 2 points cannot be shortest. This fact prejudices many people to expect the mass along the straight path to reach the bottom first.

The result that we see here is quite surprising. The mass on the curved path wins the race. How is this possible when the distance travelled is more along the curved path than along the straight path?
The path of shortest distance is not always the path of shortest time.

The mass on the curved path is certainly covering a larger distance but it is quicker than the mass on straight path.

It is because the curved path here is a part of a cycloid, the curve that would be traced by a point on a circle rolling on a straight line.

Newton showed that of all the paths connecting two points at different vertical heights, motion along the cycloidal path would take the least time provided the motion is aided by gravity alone.
The cycloidal path is more steeper in the beginning than the straight path.

As the masses move down along the 2 paths, they lose their potential energy which gets transformed into their kinetic energies.

Since the total energy is conserved, the mass which loses more amount of potential energy will gain a larger kinetic energy.

Since the cycloidal path is more steeper in the beginning than the straight path, the mass on cycloidal path gains more kinetic energy. That is, it attains a larger velocity sooner than the  mass on the straight path and also has a larger inertia of motion because of which it manages to win the race even though the last section of the path is almost flat.