The physics of parasailing Dr Andrew French. August 2013

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 1961 Pierre-Marcel Lemoigne develops the first ascending-gliding parachute as a training tool for parachutists. Cheaper than using an aeroplane! 1974 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. http://www.parasail.org/ http://en.wikipedia.org/wiki/Parasailing

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

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

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

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

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

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

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! http://en.wikipedia.org/wiki/Drag_coefficient

Reference data: Lift to drag ratio http://en.wikipedia.org/wiki/Lift-to-drag_ratio