Lissajous Curves: Roller Coasters

Ground level is given by the line y = −50. Distances are in metres and t is measured in seconds. Customers ride an elevator to the starting point (0,50) where the ride train starts and finishes.

We can then work out the speed and acceleration for this ride:

This therefore gives a speed of:

We can plot this to find the maximum speed:

This has a maximum when t = 1.053 seconds, with a maximum speed of over 156 m/s.

We can also work out the acceleration:

This gives:

Plotting this we can see the maximum acceleration:

This has a maximum acceleration of 452 m/s^2.  

Is this a safe ride? 

We can see from safety guidelines that an acceleration of 4-6g is within human range.  This would be an acceleration of up to around 60 m/s^2.  So our ride is definitely not safe for people!

How to make this ride safe?

The question can become, what is the value of a in the following vector such that the ride has a maximum acceleration of 60 m/s^2 or less?

So we first find the acceleration as:

This gives:

We can then take out a as a factor:

We can then see that the maximum of the root is given by:

Therefore we can find the maximum value of a by:

This gives the following graph:

This then gives the following rollercoaster:

It will have a height of around 13 meters and the ride will complete one loop after 2pi seconds (about 6.3 seconds).  The maximum acceleration will be around 60 m/s^2.  Play around to see what roller coaster you can create!