You would think that if every paddler could produce the same amount of power, then 20 paddlers would provide 20 times more power than one paddler. Right?
Yes, you are right.
You would also think that 20 paddlers should be able to make the boat go 20 times faster than one paddler in the same dragon boat. Right?
Water is to blame. Water resists things moving through it, including boats. The faster you move the more it resists – the more power you have to provide.
It all pivots around the drag formula:
The formula below will show you that the Power requirements go up as the cube of the speed (velocity).
Given the following:
p = fluid density (salt water, fresh water, hot water, cold water, air, oil, etc.)
C = drag coefficient (how streamlined the boat hull)
A = cross sectional area (the area of boat hull pushing through the water)
v = relative velocity (difference between the speed of the boat and the water)
k = (½ x C x p x A). The shape of the boat and the water characteristics stay the same, they equate to a certain number (a constant). And I will assume the constant is 1. That way the answer is a Power Factor rather than the actual power.
Drag = ½ x C x p x A x v2
Power = Force x v
and since Drag is a force
Power = Drag x v
Power = (½ x C x p x A x v2) x v
= (½ x C x p x A) x v3
= k x v3
Power Factor = v3
The Power Factor simply shows the overall relationship between Power and Velocity.
Suppose one paddler, George, has enough power to move the boat by himself at 5 km/hr.
The power required to do that depends on a whole bunch of things including the shape and texture of the boat, the type of water (salt, fresh, hot, cold), and the speed.
All of these variables affect the drag experienced by the boat. And it is the paddling power required to match this drag that limits the speed of the boat. The kicker, as we have seen, is that the power required to work against this drag is proportional to the speed CUBED. Yes cubed!
So in this case, for George, to move the boat at 5 km/hr requires a power factor* of 53 = 125. (Power is proportional to velocity cubed)
The power factor required to double the speed (2 x 5 = 10km/hr) is (2×5)3 = 1000 = 8 x 125
8 times more power to go 2 times the speed. Ouch!
Put another way (if all the paddlers had the same power as George):
- 1 paddler with a Power Factor of 125 could move a boat at 5km/hr
- We would need 8 paddlers with a combined Power Factor of 1000 in order to go 10km/hr (2 times faster).
- We would need 27 paddlers with a combined Power Factor of 3375 to go 15km/hr (3 times faster)
- To go 20 times faster (100km/hr), we would need 8000 paddlers like George!! Ridiculous I know.
How Fast Can the 20s Boat Go?
OK, so a 20 paddlers cannot go 20 times as fast as one paddler.
So how fast CAN the boat go with 20 paddlers each with the same power as George?
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