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 arenow possible.

http://en.wikipedia.org/wiki/Parasailinghttp://www.parasail.org/

v

y

x

LF

2F

mg2T

1T

1TMg

1Fg

gv1T

M

1F2Tm

2F

LFGravitational field strength 9.81ms-2

Velocity of tow boat /ms-1

Tension in tow cable /N

Mass of passenger + harness /kg

Drag experienced by passenger /N

Overall tension in parachute cables /N

Mass of parachute /kg

Drag experienced by parachute /N

Lift experienced by parachute /N

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

2T

Let us apply Newton’s second law in x and ydirections to the passenger and the parachute

LFmgT

FT

TMgT

TFT

sin0

cos0

sinsin0

coscos0

2

22

21

211

Passenger

Parachute

x

y

x

y

[1]

[2]

[3]

[4]

Let us parameterize the lift and drag forces as follows.

211 vkF

222 vkF

2vkF LL

Dck 121

1 2

221

2 Rck 2

21 Rck LL

Cross sectional area of passenger

Radius of parachute

Drag coefficients

Lift coefficient Density of air

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

2

2

222

21

22

11

sin

cos

sinsin

coscos

vkmgT

vkT

TMgT

TvkT

L

Parachute

x

y

x

y

[1]

[2]

[3]

[4]

Passenger

cos

sintan

22

1

2

Tvk

TMg

1tan

22

21

22

22

2

vkvk

vkgmM

vkvkL

21

22

2

1tan

kk

vkgmM

kkL

tan1 212

2

kkkk

kgmMv

L

or

[2]/[1]

22

21

2

tanvkvk

mgMgvkL

Hence

2

2

21

22

2

tan

tan

vk

vkmg

vk

vkmg

L

L

2

2

222

21

22

11

sin

cos

sinsin

coscos

vkmgT

vkT

TMgT

TvkT

L

sin

sin21

TMgT

cos

22

2

vkT

We can now work out the tensions

21

22

2

1tan

kk

vkgmM

kkL

tan1 212

2

kkkk

kgmMv

L

[4]/[3]

Now

[1][2]

[3]

[4]

Also since v must be a real quantity

tan1 212

2

kkkk

kgmMv

L

tan1

0tan1

21

2

212

kk

kk

kkkk

L

L

Hence

21

21

1tan

kk

kkL

2

222

0

v

gmMk

vk

gmM

k

k

L

L

Lk

gmMv

Now since must be positive

21

22

2

1tan

kk

vkgmM

kkL

In summary:

Some typical values .....

v = 0 .... 5ms-1 (Note 1ms-1 = 1.944 knots)

= 30o ..... 70o

c1 = 1 c2 = 1 cL = 10

= 1kgm-3

R = 4mM = 80kgm = 10kg

Lk

gmMv

22

21tan

vk

vkmg L

sin

sin21

TMgT

cos

22

2

vkT

21

21

1tan

kk

kkL

tan1 212

2

kkkk

kgmMv

L

Dck 121

1 2

221

2 Rck 2

21 Rck LL

22

121

22

Rc

Dckk

cckk LL

Very much a guess!

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

Reference data: Drag coefficient

These results indicate I may have beentoo conservative in setting the parasaildrag coefficient to be 1.

Perhaps c2 = 0.4 and cL = 4 would be moreappropriate (while keeping the lift : drag ratio as ten).Further information is needed!

http://en.wikipedia.org/wiki/Lift-to-drag_ratio

Reference data: Lift to drag ratio