Boat Math
This could get complex.
But we're not going to let it. Algebra, well, yes. We're not just balancing a check book here. Square roots. Yep. But no trigonometry. And definitely no calculus. So, if we've already scared you with the algebra and square root stuff, well, bye, bye. The rest of you, stick with us. There may be some useful things here.Theoretical Maximum Speed of a Displacement Hull:
Meaning the Maximum Hull Speed in Knots of a displacement hull boat is equal to 1.34 times the square root of the waterline length.
So, if, for instance, your waterline length is (conveniently) 36 feet, your Maximum Hull speed is 1.34 times 6 (6, of course, being the square root of 36) which equals 8.04 knots - roughly 8 knots.
Why is this? It has to do with the bow wave a displacement hull creates as it plows through the water. Once the crest to crest distance of the bow wave is equal to the length of the boat, the boat cannot go any faster without running over the wave which it is creating - which would be tantamount to getting up on a plane, but it is a displacement hull, not a planing hull. So, by definition, this is the maximum speed.
There are stories we've heard of the clipper ships which to gain maximum speed, they had so many sails that they passed their own bow wave, This plunged the bow of the boat down into the water and caused them to sink - literally submarining. We cannot verify these stories.
So, if, for instance, your waterline length is (conveniently) 36 feet, your Maximum Hull speed is 1.34 times 6 (6, of course, being the square root of 36) which equals 8.04 knots - roughly 8 knots.
Why is this? It has to do with the bow wave a displacement hull creates as it plows through the water. Once the crest to crest distance of the bow wave is equal to the length of the boat, the boat cannot go any faster without running over the wave which it is creating - which would be tantamount to getting up on a plane, but it is a displacement hull, not a planing hull. So, by definition, this is the maximum speed.
There are stories we've heard of the clipper ships which to gain maximum speed, they had so many sails that they passed their own bow wave, This plunged the bow of the boat down into the water and caused them to sink - literally submarining. We cannot verify these stories.
Calculating most efficient Cruising Speed
This one isn't quite as cut and dried, put down the formula and plug in the numbers kind of math. There are a lot of variables. It helps if you have a 'flow meter' for your fuel line.
First, you need to find out what your fuel usage is at various engine RPM's. A flow meter makes this a lot easier. Not having that, you need to run the engine at a constant RPM after having topped off your tank. And you have to do it long enough to minimize the percent of the time you run it at different speeds while you are docking or maneuvering. This could be days or weeks if you don't travel much.
So you top off your tank, write down the engine hours. Then every day while cruising, you run the engine at the same speed. Once you have an additional 20 or 30 engine hours, top off the tank again. And write down the amount of fuel used.
Say you have gone 30 hours and used 25 gallons of fuel. Divide the amount of fuel used by the number of hours (25 / 30 = .83 gallons per hour in our example.)
Start your chart:
First, you need to find out what your fuel usage is at various engine RPM's. A flow meter makes this a lot easier. Not having that, you need to run the engine at a constant RPM after having topped off your tank. And you have to do it long enough to minimize the percent of the time you run it at different speeds while you are docking or maneuvering. This could be days or weeks if you don't travel much.
So you top off your tank, write down the engine hours. Then every day while cruising, you run the engine at the same speed. Once you have an additional 20 or 30 engine hours, top off the tank again. And write down the amount of fuel used.
Say you have gone 30 hours and used 25 gallons of fuel. Divide the amount of fuel used by the number of hours (25 / 30 = .83 gallons per hour in our example.)
Start your chart:
| RPM | GPH | SPEED | GPM | MPG |
| 1500 | .83 | |||
Continue over time adding to your chart, as below:
(that flow meter is beginning to sound like a pretty good idea, isn't it.)
(that flow meter is beginning to sound like a pretty good idea, isn't it.)
| RPM | GPH | SPEED | GPM | MPG |
| 1500 | .83 | |||
| 1800 | .97 | |||
| 2000 | 1.1 | |||
| 2200 | 1.34 | |||
| 2500 | 1.9 |
Yes, I know we haven't filled in the speed yet. And there's a reason. Now you have to figure out what your speed is a varying RPM's. And because there are wind and current variations, you need to run your boat in two directions. So measure out a mile, or so on your chart plotter or with your GPS. Get the boat up to speed at each RPM and take the average in each direction. Say, for instance, it takes you 13 minutes to run your mile in one direction and 11 minutes in the other. That's an average of 12 minutes per mile. In an hour then, you would have traveled 5 miles (or nautical miles - whichever you're calculating.) Write that down on your chart and do it again at the next RPM on your list.
| RPM | GPH | SPEED | GPM | MPG |
| 1500 | .83 | 4.9 | ||
| 1800 | .97 | 5.7 | ||
| 2000 | 1.1 | 6.6 | ||
| 2200 | 1.34 | 7.0 | ||
| 2500 | 1.9 | 7.3 |
So at 1500 it takes an hour to go 4.9 miles and you use .83 gallons. That's .166 gallons per mile. Or 4.15 miles per gallon. Gallons per Mile is GPH / Speed. Miles per Gallon is Speed / GPH.
| RPM | GPH | SPEED | GPM | MPG |
| 1500 | .83 | 4.9 | .169 | 5.90 |
| 1800 | .97 | 5.7 | .170 | 5.87 |
| 2000 | 1.1 | 6.6 | .166 | 6.00 |
| 2200 | 1.34 | 7.0 | .191 | 5.22 |
| 2500 | 1.9 | 7.3 | .260 | 3.84 |
As you can see from this chart, about 2000 RPM gives you your most efficient speed. We made these figures up, But most of the time you will find figures proportional to these. Somewhere in the middle of your engine's RPM range .
We like the Gallons per mile column, too, because if we're going to be on a long cruise, for instance, up or down the ICW we can calculate about how much fuel will cost. So if the cruise is going to be, say 1200 miles we are going to use 199.2 gallons (1.66 x 1200). So figure about 200 gallons. If fuel is averaging $3.75 / gallon, we need to budget $750.00 for fuel Currents and winds tend to cancel each other out. Some days (or hours) you have them with you, the next against. Of course, if you can raise the sail, you can save some money. When you get there you can treat yourself to a dinner out on what you've saved.
If you're going to go to all the trouble to figure out your fuel efficiency, remember to keep your boat's bottom and the prop clean. And dragging your dinghy behind the boat is also a huge detriment to efficiency. Hang it from davits or put it on deck, hopefully somewhere where it won't obstruct your view.
We like the Gallons per mile column, too, because if we're going to be on a long cruise, for instance, up or down the ICW we can calculate about how much fuel will cost. So if the cruise is going to be, say 1200 miles we are going to use 199.2 gallons (1.66 x 1200). So figure about 200 gallons. If fuel is averaging $3.75 / gallon, we need to budget $750.00 for fuel Currents and winds tend to cancel each other out. Some days (or hours) you have them with you, the next against. Of course, if you can raise the sail, you can save some money. When you get there you can treat yourself to a dinner out on what you've saved.
If you're going to go to all the trouble to figure out your fuel efficiency, remember to keep your boat's bottom and the prop clean. And dragging your dinghy behind the boat is also a huge detriment to efficiency. Hang it from davits or put it on deck, hopefully somewhere where it won't obstruct your view.
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Calculate your Boat's Speed
So your speed log is clogged up and your GPS battery is dead, and suddenly, without any warning it becomes imperative to calculate your boat's speed, here's how you do it:
Throw something that floats in the water at the front of your boat and time how long it takes to get to the back of your boat. (Tie a string to retrieve your floating object so you don't pollute.) (And don't let the string foul your prop.)
Or time how long it takes to pass a buoy, or some other object in the water. Now you have a couple of variables to work with - time in seconds and the length of your boat in feet. Plug those variables into this formula:
S= 3600 X L / 6076.1 X T
where S = Boat Speed in Knots
L = Length of your boat in feet
T = Time in seconds it takes for you to pass the buoy or your floating object to pass.
The constants: 3600 is the number of seconds in an hour and
6076.1 is the number of feet in a nautical mile.
You can probably round off - this isn't going to be that exact.
Example: Your boat is 37 feet long. It takes 6 seconds to pass.
S = 3600 X 37 / 6076 X 6
S = 133200 / 36456
S = 3.653 knots
Figure you're doing a little over 3 1/2 knots.
Hang on to your hat!
Throw something that floats in the water at the front of your boat and time how long it takes to get to the back of your boat. (Tie a string to retrieve your floating object so you don't pollute.) (And don't let the string foul your prop.)
Or time how long it takes to pass a buoy, or some other object in the water. Now you have a couple of variables to work with - time in seconds and the length of your boat in feet. Plug those variables into this formula:
S= 3600 X L / 6076.1 X T
where S = Boat Speed in Knots
L = Length of your boat in feet
T = Time in seconds it takes for you to pass the buoy or your floating object to pass.
The constants: 3600 is the number of seconds in an hour and
6076.1 is the number of feet in a nautical mile.
You can probably round off - this isn't going to be that exact.
Example: Your boat is 37 feet long. It takes 6 seconds to pass.
S = 3600 X 37 / 6076 X 6
S = 133200 / 36456
S = 3.653 knots
Figure you're doing a little over 3 1/2 knots.
Hang on to your hat!
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Rule of Twelfths in Tide Prediction
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Calculating Theoretical Range of your VHF radio from antenna Height
Range in Nautical Miles; Antenna Height in Feet
So if your antenna height is 55 feet, your range to the horizon is ~9 1/8 Nautical miles
So your math teacher says, "Show your work!"
1.23 x square root of 55
1.23 x 7.416
9.12 or about 9 1/8 nautical miles
Of course the other guy to whom you're transmitting also has his antenna somewhere above the horizon, unless he's laying down in his dinghy. So you can add his possible range to yours. So if his antenna is, say, 25 feet in the air his range is 6.15 miles, so you can talk to this guy until you're a little over 15 nautical miles from him. Unless there's something in the way, or you're operating on low power, or the atmospheric conditions aren't right. Lots of holes in this theory.
So if your antenna height is 55 feet, your range to the horizon is ~9 1/8 Nautical miles
So your math teacher says, "Show your work!"
1.23 x square root of 55
1.23 x 7.416
9.12 or about 9 1/8 nautical miles
Of course the other guy to whom you're transmitting also has his antenna somewhere above the horizon, unless he's laying down in his dinghy. So you can add his possible range to yours. So if his antenna is, say, 25 feet in the air his range is 6.15 miles, so you can talk to this guy until you're a little over 15 nautical miles from him. Unless there's something in the way, or you're operating on low power, or the atmospheric conditions aren't right. Lots of holes in this theory.
This little math formula, actually more like a table, is useful for calculating heights of the tide based on time of slack and tidal range in areas of diurnal (twice a day) tides. This is useful if you are trying to time your approach to a fixed bridge which, at high tide is too low for the height of your mast or a shallow area, which you need to pass over at some point other than low tide.
Slack
+1 Hour - 1 / 12 of range
+2 Hours - 2 / 12 of range
+3 Hours - 3 / 12 of range
+4 Hours - 3 / 12 of Range
+ 5 Hours - 2 / 12 of Range
+ 6 Hours - 1 / 12 of Range
Slack
+1 Hour - 1 / 12 of range
+2 Hours - 2 / 12 of range
+3 Hours - 3 / 12 of range
+4 Hours - 3 / 12 of Range
+ 5 Hours - 2 / 12 of Range
+ 6 Hours - 1 / 12 of Range
Slack
So let's give a 'for instance':
Let's say you're in Georgia traveling down the ICW. The tidal range is 8 feet. The top of your mast is 66 feet. At mean high the bridge clearance is 63 feet.
Low tide is at 7:00 AM. So at low tide you have 71 feet clearance. You have 5 feet to play with.
At 8:00 AM. The tide has already come up about 2 / 3 of a foot. (1 / 12 X 8 ). So you have a little over 4 feet above your mast.
At 9:00 AM it's going to be an addition 1 1/3 feet ( 2/12 X 8 ) and you add that to the 2 / 3 foot you had in the first hour so you have 2 feet of tide has come in. Now you have a bridge clearance of 69 feet so you have 3 feet above your mast.
By 10:AM you add another 2 feet to the incoming tide. Bridge clearance is 67 feet. It's getting tight. That scant foot to play with would make me pretty nervous. There could be other factors; maybe it's going to be a high high, not a mean high tide. Or there could be wind driven current which may add to the problem. Personally, I'd say, if you can't get to that bridge well before 10:00 AM, wait for the tide to be back on its way out.
Make your own little chart while you're sitting at anchor, sipping your sundowner the evening before so you know what time you need to get up to get under that bridge,
Let's say you're in Georgia traveling down the ICW. The tidal range is 8 feet. The top of your mast is 66 feet. At mean high the bridge clearance is 63 feet.
Low tide is at 7:00 AM. So at low tide you have 71 feet clearance. You have 5 feet to play with.
At 8:00 AM. The tide has already come up about 2 / 3 of a foot. (1 / 12 X 8 ). So you have a little over 4 feet above your mast.
At 9:00 AM it's going to be an addition 1 1/3 feet ( 2/12 X 8 ) and you add that to the 2 / 3 foot you had in the first hour so you have 2 feet of tide has come in. Now you have a bridge clearance of 69 feet so you have 3 feet above your mast.
By 10:AM you add another 2 feet to the incoming tide. Bridge clearance is 67 feet. It's getting tight. That scant foot to play with would make me pretty nervous. There could be other factors; maybe it's going to be a high high, not a mean high tide. Or there could be wind driven current which may add to the problem. Personally, I'd say, if you can't get to that bridge well before 10:00 AM, wait for the tide to be back on its way out.
Make your own little chart while you're sitting at anchor, sipping your sundowner the evening before so you know what time you need to get up to get under that bridge,
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We'll try to have more Boat Math for you Soon.
We'll try to have more Boat Math for you Soon.
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