There is a physics problem called the "Brachistochrone problem" which seeks to identify the "path of quickest descent" for a frictionless body on a ramp with the only force acting being gravity.
For bodies with 0 initial velocity, the problem was solved centuries ago by Isaac Newton.
The problem is, due to friction, without an initial velocity, there is no possibility of descending a ramp and rising back to the initial height, where the start and end points are horizontal to each other.
To start and end on the same height, one needs an initial velocity, but that is precisely where things get complicated.
It turns out the solution for a given velocity is "irrational" meaning it would take the fastest computer an infinite amount of time to solve precisely, but it can be solved "for all practical purposes" in the blink of an eye.
The quickest path between 2 horizontal points with an initial velocity is a portion of a curve known as a "cycloid." The cycloid can be thought of as the path that a point on the edge of a wheel takes as the wheel rolls along the ground. In order to identify the portion of the cycloid that is the quickest path between 2 points with an initial velocity, one must remove the "cusps" of the cycloid such that a free-falling body would reach the initial velocity after falling that vertical height, and the new, truncated cusps must traverse the desired horizontal distance.
To find the curve, one must identify 2 variables: the cycloid generating circle radius, and the initial angle.
It turns out the initial angle is irrational but it can be worked out to arbitrary precision with numerical computation methods as there is no algebraic solution.
Where A = -0.05102 m, the depth that accelerates objects to 1m/s in 9.8m/s^2 gravity), and B is the horizontal distance (B = 1 meter):
A=-(B/2(pi-t+sin(t)))(1-cos(t))
-0.05102=-(B/2(pi-t+sin(t)))(1-cos(t))
therefore:
t=0.8117451055281
Therefore the initial angle of the cycloid is 0.8117451055281 radians and the final angle is 2pi-0.8117451055281 radians.
Next, the cycloid generating radius is found via:
r=B/(2(pi-t+sin(t)))
Therefore:
r=0.1636480760880076136181 meters
Parametric Graph: parametric graphing
Next we find the travel time of the whole cycloid without the cusps removed with 0 initial velocity:
time_whole_cycloid=2*pi*sqrt(r/g)
0.81193690616139705994 seconds = 2*pi*sqrt(0.1636480760880076136181/9.8)
Next we find the travel time that it takes to accelerate from the cusps with 0 initial velocity to the desired initial velocity:
dtime=sqrt(a/g)*dtheta
dtime = sqrt(0.1636480760880076136181/9.8)* 0.8117451055281
0.1048967645154483811775 seconds = sqrt(0.1636480760880076136181/9.8)* 0.8117451055281
Next we subtract the time it takes to accelerate to and from the initial velocity from the time it takes to traverse the un-truncated cycloid:
0.602143377130500297585=0.81193690616139705994-0.1048967645154483811775-0.1048967645154483811775
Giving an answer of:
0.602 seconds to traverse the quickest path between 2 horizontal points separated by 1 meter with an initial velocity of 1 meter per second.
There is a more lengthy discussion on the topic here:
Quickest Route
Downloadable Excel Spreadsheet
quickest_route - Shared with pCloud
As a consequence of the fact that a distant horizontal point can be reached in less time than the straight path with the initial velocity, huge energy savings or performance increases are possible.
For example, using the spreadsheet linked above, I find the following results:
In summary, a trip between 2 horizontal points separated by 5km with an initial velocity of 9 meters per second, takes 10.1x longer going in a straight line than on the optimal curved path.
For bodies with 0 initial velocity, the problem was solved centuries ago by Isaac Newton.
The problem is, due to friction, without an initial velocity, there is no possibility of descending a ramp and rising back to the initial height, where the start and end points are horizontal to each other.
To start and end on the same height, one needs an initial velocity, but that is precisely where things get complicated.
It turns out the solution for a given velocity is "irrational" meaning it would take the fastest computer an infinite amount of time to solve precisely, but it can be solved "for all practical purposes" in the blink of an eye.
The quickest path between 2 horizontal points with an initial velocity is a portion of a curve known as a "cycloid." The cycloid can be thought of as the path that a point on the edge of a wheel takes as the wheel rolls along the ground. In order to identify the portion of the cycloid that is the quickest path between 2 points with an initial velocity, one must remove the "cusps" of the cycloid such that a free-falling body would reach the initial velocity after falling that vertical height, and the new, truncated cusps must traverse the desired horizontal distance.
To find the curve, one must identify 2 variables: the cycloid generating circle radius, and the initial angle.
It turns out the initial angle is irrational but it can be worked out to arbitrary precision with numerical computation methods as there is no algebraic solution.
Where A = -0.05102 m, the depth that accelerates objects to 1m/s in 9.8m/s^2 gravity), and B is the horizontal distance (B = 1 meter):
A=-(B/2(pi-t+sin(t)))(1-cos(t))
-0.05102=-(B/2(pi-t+sin(t)))(1-cos(t))
therefore:
t=0.8117451055281
Therefore the initial angle of the cycloid is 0.8117451055281 radians and the final angle is 2pi-0.8117451055281 radians.
Next, the cycloid generating radius is found via:
r=B/(2(pi-t+sin(t)))
Therefore:
r=0.1636480760880076136181 meters
Parametric Graph: parametric graphing
Next we find the travel time of the whole cycloid without the cusps removed with 0 initial velocity:
time_whole_cycloid=2*pi*sqrt(r/g)
0.81193690616139705994 seconds = 2*pi*sqrt(0.1636480760880076136181/9.8)
Next we find the travel time that it takes to accelerate from the cusps with 0 initial velocity to the desired initial velocity:
dtime=sqrt(a/g)*dtheta
dtime = sqrt(0.1636480760880076136181/9.8)* 0.8117451055281
0.1048967645154483811775 seconds = sqrt(0.1636480760880076136181/9.8)* 0.8117451055281
Next we subtract the time it takes to accelerate to and from the initial velocity from the time it takes to traverse the un-truncated cycloid:
0.602143377130500297585=0.81193690616139705994-0.1048967645154483811775-0.1048967645154483811775
Giving an answer of:
0.602 seconds to traverse the quickest path between 2 horizontal points separated by 1 meter with an initial velocity of 1 meter per second.
There is a more lengthy discussion on the topic here:
Quickest Route
Downloadable Excel Spreadsheet
quickest_route - Shared with pCloud
As a consequence of the fact that a distant horizontal point can be reached in less time than the straight path with the initial velocity, huge energy savings or performance increases are possible.
For example, using the spreadsheet linked above, I find the following results:
In summary, a trip between 2 horizontal points separated by 5km with an initial velocity of 9 meters per second, takes 10.1x longer going in a straight line than on the optimal curved path.
