1. The physics of parasailing
Dr Andrew French.
August 2013

2. What is parasailing? Parasailing (or parascending, parakiting)
is an activity whereby a passenger is lofted into the air via a modified parachute (called a parasail wing) towed by a motorized vehicle. For safety reasons, recreational parasailing is typically offered as a watersport.
History
Pierre-Marcel Lemoigne develops the first ascending-gliding parachute as a training tool for parachutists. Cheaper than using an aeroplane!
Mark McCulloh invents the basic parasail launch and recovery system (“Winchboat”) via a cable winch system. This setup is the basis of modern commercial operations.
1976 Brian Gaskin created the ‘16-gore’ waterproof canopy design (“Waterbird”) and passenger harness.
1990s+
Much larger canopies designed with higher
lift : drag ratios. Multiple passenger rides are
now possible.

3. A mathematical model of parasailing
Once the tow cable has been deployed, the
cable and parachute cord angles are observed to be constant for a given boat velocity.
One shall therefore consider the entire system to be in dynamic equilibrium i.e. there is no net force or consequential acceleration.
Physical parameters
Lift experienced by parachute /N
Gravitational field strength 9.81ms-2
Drag experienced by passenger /N
Velocity of tow boat /ms-1
Overall tension in parachute cables /N
Tension in tow cable /N
Mass of parachute /kg
Mass of passenger + harness /kg
Drag experienced by parachute /N

4. Let us apply Newton’s second law in x and y
directions to the passenger and the parachute
Passenger x
[1] y
[2]
Parachute
This is the potentially dodgy bit of the analysis! The drag and lift forces will change with angle of attack. However, in the absence of a data sheet we have got to start somewhere!
As a first approximation set drag and lift coefficients c2 and cL to be constants. i.e. independent of angle q + f x
[3] y
[4]
Let us parameterize the lift and drag forces as follows.
Drag coefficients
Cross sectional area of passenger
Radius of parachute
Lift coefficient
Density of air

5. Passenger x
[1] y
[2]
[2]/[1]
Parachute
[3] x
[4] y
Hence or

6. Now
[1]
[4]/[3]
[2]
[3]
[4]
We can now work out the tensions

7. Now since q must be positive
Also since v must be a real quantity
Hence

8. v = 0 .... 5ms-1 (Note 1ms-1 = 1.944 knots) q = 30o ..... 70o
In summary:
Some typical values .....
v = ms-1 (Note 1ms-1 = knots)
q = 30o o
r = 1kgm-3
R = 4m
M = 80kg
m = 10kg
c1 = 1
c2 = 1
cL = 10
Very much a guess!

10. Reference data: Drag coefficient
These results indicate I may have been
too conservative in setting the parasail
drag coefficient to be 1.
Perhaps c2 = 0.4 and cL = 4 would be more
appropriate (while keeping the lift : drag ratio as ten).
Further information is needed!

11. Reference data: Lift to drag ratio