# Some dragonboat paddling physics: force mass and rate

This post arose due to an attempt to understand why one boat with apparent equal strength is faster than another. What resulted was an investigation into how applied paddling pressure and boat mass affects variables such as paddle speed, effective acceleration.

This was originally an email draft to the coach group, but as it became rather wordy, I felt it would be better served as a blog post.

Just for illustration’s sake, we’ll simplify the numbers, and strip out the physics calculations. We’ll just make some assumptions to simply this too:

• Assuming all paddlers in perfect rhythm.
• Assuming 20 paddlers in total
• Assuming total boat-load is 2000kg (approx 250kg + 22 x 80kg, including drummer & helm).
• Average mass each paddler has to move: 2000/20 = 100kg.

Lets just also assume a paddler can exert 10N of force. 10kg at 1m/s².
Also to simplify calculations even more, the paddlers each pull a 1m path in the water, and that the force exerted is equal along this path.

Most of these are already intuitive from experience in the water, but here, understanding exactly how the forces are being applied is not as obvious. We’ll not cover drag from the paddles and from the water on the boat for now.

#### 1st intuitive statement: Each application of force (re: paddle stroke) adds to the momentum of the boat.

So at standstill, the paddle will not initially move much relative to the boat (acceleration = Force / mass; force is the total pressure on point in time, and as the boat mass is typically much larger than the paddler’s applied force, the resultant acceleration is tiny for one person).

When the boat is moving at speed, the paddle when buried in water will move at the same speed as the boat speed relative to the water without applying additional pressure. If you then apply more pressure, this will cause the paddle to accelerate faster than the boat relative to the water. This difference can be determined by the above equation to add to the resultant velocity.

e.g. 10N / 100kg = 0.1 m/s^2.

We’ll just assume for now that power application from the paddler is the same against static pressure (like a wall), and versus dynamic pressure (in practise, this uses different types of muscle fibres), and the power graph is linear.

Just some notes to the non-physics people:

• pressure is force applied per area (N / sqm) – this relates to say, different paddle sizes where you can apply the same force, but the resultant movement of the paddle is different due to the pressure being spread larger over a larger blade.
• Force is Mass applied with acceleration (e.g. the application.of a moving a cannonball weighing 20kg to accelerate at 1m/s^2) – this is the exertion at a point in time.
• Work is Force over Time (W =Fs), and we use this to state total effort.
• s^2 is just another way of writing seconds squared (s²).
##### Application of stroke rate

How does effective acceleration relate to the stroke rate? The rate is affected by these parameters:

1. length of effective stroke in the water (m)
2. speed of recovery – the time it takes for the paddle to get back into position (s)
3. how long it takes for the paddle blade to be applied in the water (s)

So assuming the first two are constant for now: the path of the paddle is stays at the same length, and recovery speed stays constant.  How long the paddle stays in the water is related to how much force is being applied over the length of the stroke, which is effective work done: Work = Force x distance = the total energy expended.

#### Intuitive statement #2: When more pressure is applied, the paddle can pull the boat harder, and the boat speed goes up.

If the stroke path is lengthened, and assuming the same force application, then total work performed increases. So the inverse when the path is shortened is that less work is performed (funnily enough), and that means the boat is given lower overall acceleration.

#### Intuitive statement #3: Higher acceleration results in a longer travelled distance

Note that the maximum length of the paddle stroke is due to bio-mechanical limits – this site actually has a very good read on the process of the stroke:

Just to see how less paddlers affect boat acceleration, lets assume 10+2 people on the same boat:

• Total mass = 250 + 12*80 = 1210
• Average mass to move per paddler: 1210/10 = 121kg.

Remembering a = F/m, this states that the boat will accelerate slower with the same applied force in the water – because m is bigger, the resultant acceleration is smaller.

`a = 10N / 121 = 0.083 m/s^2`

This means the resultant stroke rate goes down as the paddle needs more time in the water to travel the same length as before.

Remember, to maintain the same amount of total Acceleration to move the boat equally, you would need a higher force per paddler (approx 20% more effort in this case) to expend higher pressure for the same path length over the same time period (see point iii above).

The conclusion with this is that higher mass to move per paddler = lower effective stroke rate, with all other aspects being equal.

#### Effects of a higher rate with less paddlers

So keeping this in mind, if a higher rate is being applied with less paddlers on the boat, what could cause this change? Possibly:

• faster recovery (less time [s], so more of the stroke is dedicated to being in the water)
• shorter stroke (less work performed)
• higher power application (maintaining work)

If faster recovery does not affect the stroke length, then under perfect circumstances you can expend more time to the pull phrase. There are two ways to minimise the recovery period:

1. higher recovery velocity to maintain same recovery length for the setup
2. same recovery velocity, so shorter recovery length, leading to shorter stroke.

There’s a triangle of dependencies between stroke length, recovery time, and recovery length – the only way to maintain full power output here is keeping stroke length and recovery length – but minimize recovery time.

Minimising recovery comes with its own caveats in case 1: forcing muscles to work when recovering means the muscle’s oxygen & energy reserves are depleted quicker, leading to higher anaerobic activity, so this burst of activity can only last a limited time.

Just to state the flip side, a faster rate with a shorter stroke works if the total effective work is higher – e.g. 100% stroke efficiency at 60 spm vs 51% stroke efficiency at 120 spm.

Typically a higher-rate stroke will have various compromises of the aforementioned changes with a shorter stroke, higher force and faster recovery, with the aim that this will be enough to produce a higher boat velocity.

#### Comparison with a lighter boat

It’s been mentioned in the past that Typhoon with our smaller paddlers should be using a higher rate.

What’s behind this idea and how?

Let’s assume that everyone on the boat is 50kg (yes a very light crew!).

• Let’s say the force output per person is 25% less, at 7.5N.
• Total boat-load is 1350kg (250kg + 22 x 50kg).
• Average mass each paddler has to move: 1350/20 = 67,5kg.

Using a=F/m and the equation for uniform acceleration, t=sqr(2s/a) to calculate the time to travel 1 metre:

Acceleration: 7.5/67.5 = 0.11 m/s^2

With the initial stroke, this would take sqr(1*2 / 0.11) = sqr(18) = 4.24 seconds to travel 1 metre.

Compare this against the other boat scenarios to travel the first  metre:

• 20 paddlers + helm + drummer at 80kg applying 10N each: sqr(1*2 / 0.1) = 4.47 seconds
• 10 paddlers + helm + drummer at 80kg applying 10N each: sqr(1*2 / 0.083) = 4.91 seconds

We can say that the lighter crew has the capability to paddle at a higher rate (however this is assuming that their power output is not proportionally lower).

Alternatively, we can flip this around and look at how long the lighter paddlers need to accelerate through the water to achieve the same velocity increase per stroke as the 80kg boat (using v = at):

• Original boat of 20+1+1 x 80kg paddlers: v= 0.1 * 4.47 = 0.447 m/s velocity (1.6 kmph in the first 1m stroke)

We then use this to calculate the needed time interval to achieve 0.447 m/s:

• Light boat: t = 0.447 / 0.11 = 4.06 seconds
• 10 paddler boat: t = 0.447 / 0.083 = 5.39 seconds (for comparison purposes, assuming they can paddle longer than 1m!)

Finally, how much shorter the lighter boat needs to paddle to achieve this velocity:

`s = 0.5 at^2 = 0.5 * 0.11 * 4.06^2 = 0.916 metres`

Remember, these numbers all applies to the first stroke. Subsequent strokes will have lower time intervals, but they will follow the same rules, and the same proportions of acceleration.

##### A fairer assessment  with the light crew – and the power profile

The above scenario relies on a generous increase in power-to-weight ratio when we have a lighter crew, so just to be exact (and fair), lets reduce the resultant force proportionally to weight too:

• Each person is 50kg (total boat load=1350kg)
• Force output per person = 6.25N
• Average mass per person = 67.5kg
• Resultant acceleration = 6.25 / 67.5 = 0.093 m/s²

So having a lighter crew in itself does not result in a more advantageous crew – in fact it is slightly disadvantageous with the using the same paddler power-to-weight ratio. However, for any crew that has higher power output (or improves their power-to-weight ratio), the previous section applies where the resultant acceleration is higher.

We haven’t mentioned the stroke profile so far – we’ve assumed so far for illustrative purposes that the force output is flat. Where allowing a faster/shorter stroke would work is to do with the shape of the stroke profile:

• If the early part of the stroke produces most of the force, then it makes sense to finish the stroke earlier to make more use of the front of the stroke.
• If the force profile is relatively flat, then maximising the length should take priority.

Having non-uniform acceleration complicates calculations, so this will have to be dealt with another time.

### Conclusions

The various scenarios show that:

1. Higher power-to-weight ratios lends itself to higher acceleration;
2. Stroke rate is a result of the applied force and how quickly the boat can move past the water – if the force and boat velocity is low, the paddle will move past the boat slowly forcing a lower stroke rate; conversely, higher pressure allows the stroke rate to be higher.
3. You cannot force higher rates without increasing power, compromising length or the recovery period

The stroke profile is also quite important in determining the optimal drive length and rate based on measured power output.

One assumption we made earlier is that every paddler is the same. In practise, everyone will have a different stroke profile, and this means that the the ideal stroke/rate has to be tuned to one that allows the whole boat to contribute most efficiently together.