Between fixes the chart holds only what you work out: heading, log, leeway and stream, added as vectors.
1 · Dead reckoning
Course and distance, added as vectors
How do I convert a compass course to a true course?
Compass + deviation = magnetic, + variation = true, counting east positive: 237°C with 3°W deviation and 4°W variation is 230°T. Take the deviation from the card for that compass heading.
A DR comes from the heading steered and the distance logged. Each leg is a vector, course and distance: a change of course is a new vector, never a mean course with the summed log. The heading goes through the card for that compass heading, then the variation: compass + deviation = magnetic, + variation = true, east positive. The fluxgate reads magnetic, without the steering compass's deviation.
DR from the log book: worked example
2000 fix · 240°T, 8.0 kn to 2300: 24.0 M
2300 altered: 200°T, 8.0 kn to 0200: 24.0 M
Not 220°T, 48 M: add the vectors
0200 DR: 220°T, 45.1 M from the fix
Helm steered 237°C? Card 3°W, var 4°W: 237 − 3 − 4 = 230°T goes on the chart
Work the hourly EP from the last fix and compare it with the GNSS: a mile or two of gap in an hour is a fault; call the skipper. With the electrics dead, the magnetic compass by torchlight gives the heading, a chip timed along the deck the speed: m/s ÷ 0.514 = knots. Restart the plot from the paper log's last fix.
2 · Estimated position
Leeway and stream on top of the DR
Do I add or subtract leeway on port tack?
Add it on port tack, subtract it on starboard: leeway always turns the heading to leeward, into the water track. A motor yacht under power is allowed none.
The EP adds what the water did: leeway first, then the stream. Leeway turns the heading into the water track, always to leeward: added on port tack, subtracted on starboard, changing side when you tack. A motor yacht under power is allowed none. From GNSS, ground vector minus current is the water vector, its angle from the heading the leeway; or take the wake's bearing.
Port tackadd it
Starboard tacksubtract it
EP: worked example
0000 fix · heading 250°T · log 7.0 kn · 3 h
Wind NW: starboard tack, leeway 5° to port
Water track 245°T, 21.0 M
Streams 160° 1.0 · 165° 1.5 · 170° 1.0 kn
= 3.5 M toward 165°T
0300 EP: 236°T, 21.9 M from the fix
Sum the diamond's hourly vectors for the leg at today's rate; beyond the atlas use the routeing chart's current, checked by fixes. In heavy weather leeway grows to 15° or more; hove-to or under a drogue the drift is a vector of its own; a stopped motor yacht drifts beam-on with the wind, plus the current. Near a lee shore overstate leeway and keep sea room.
3 · Course to steer
One triangle for the whole crossing
How do I work out a course to steer across a stream lasting several hours?
Lay off the whole crossing's stream from the start as one vector, then swing an arc of speed × time through the water from its end to cut the ground track: that line is the water track to steer.
Lay off the whole crossing's stream from A, summed into one vector. From its end swing an arc of speed × time through the water to cut the ground track: that line is the water track to steer, aimed off by asin(stream across the track ÷ water distance). Stream along the track changes the hours: check them against ground distance ÷ SOG, and redo it once if they differ. Then leeway, variation, deviation.
Course to steer: worked example
Track 000°T, 41.6 M · 7.0 kn: about 6 h
Streams E 1.0 · 2.0 · 2.5 · 1.5; W 0.5 · 0.5
Net 7.0 − 1.0 = 6.0 M east
Aim off asin(6.0 ÷ 42) = 8°: steer 352°T
Leeway 4°, wind W: head up 4° more, 348°T
Var 5°W: 353°M · dev 2°E: 351°C
For a waypoint at a set time the water vector is the ground vector minus the stream over that time; its length ÷ hours is the speed to hold. To meet a vessel holding her course, keep her bearing steady: sin A = her speed × sin(her course − her bearing) ÷ yours, A off her bearing. On the same course you close at the difference of speeds.
4 · Errors and cross-track
1 in 60, and how doubt grows
How do I use the 1 in 60 rule to get back on track?
Alter by two angles from the 1 in 60 rule: miles off × 60 ÷ miles run is the track error in degrees, and the same over the miles to go is the extra turn to close.
1 in 60: one degree over 60 M is one mile off. Miles off × 60 ÷ miles run is the track error in degrees; the same sum over the miles to go is the extra turn to close. The EP's doubt grows with the run, about 10% of it; stream rate, log calibration and the helm's average course each add a share. A deviation applied backwards counts twice.
Where the error puts you
Log over-reads 4%: 104 M logged
True run 104 ÷ 1.04 = 100.0 M: 4 M short
Current 0.4 kn too weak for 12 h: 4.8 M off
Helm 2° off for 90 M: 2 × 90 ÷ 60 = 3.0 M
Used 180° 0.8 kn, real 210° 0.8 kn, 10 h: the EP is 4.1 M out, toward 285°
Cross-track error matters where the track was laid for a reason: a margin off a wind farm, a lane in a TSS. Regain the planned line, in a lane without cutting the separation zone; track mode crabs into the stream. With nothing near, steering straight for the waypoint is shortest. At a low coast aim off, so you know which way to turn.
5 · Current from fixes
Ground vector minus water vector
How do I find the current from two fixes?
Subtract the water vector, the leeway-corrected heading by the logged distance, from the ground vector between the fixes: the difference is the current, its set, and distance ÷ hours its rate.
Between two fixes the ground vector is what happened; the water vector is the leeway-corrected heading by the logged distance. The difference is the current: set, and distance ÷ hours. Stopped in a calm, the drift is the current alone. Fix minus EP sums every error since the fix, not current alone. A steady gap along the heading can be current, a log under-reading, or both.
Current from fixes: worked example
0300–0600: heading 093°T, leeway 3° to port
Water track 090°T, log run 18.0 M
Fix to fix: 20.0 M toward 100°T
Ground − water = 3.9 M toward 154°T
Over 3 h: set 154°T, rate 1.3 kn
Use it for the next leg; check at the next fix
6 · Speed made good and timing
ETA, tidal gates and daylight
What is VMG and how do I work it out when beating?
VMG is the speed at which you close the mark: boat speed × cos(water track − course to the mark), leeway included. 7.0 kn with the water track 48° off the course makes good 4.7 kn.
Speed made good is distance to go closed ÷ time over the whole run, not the mean of the legs' speeds nor the log. Beating, only VMG closes it: boat speed × cos(water track − course to the mark), leeway included; take the tack pointing nearest it. Hour by hour, SOG is boat speed plus the along-track stream.
Tidal gate and VMG: worked example
Gate 38.0 M ahead at 0900; fair stream to 1430
5.5 h: 38.0 ÷ 5.5 = 6.9 kn over the ground
Foul stream 0.8 kn: hold 7.7 kn by the log
Beating: 7.0 kn, water track 48° off course
VMG 7.0 × cos 48° = 4.7 kn
80 M to windward: 17 h 05, from 0600: 2305
Reach a tidal gate early in its fair stream, with time in hand: an ETA from the log speed, an uncalibrated log or spring rates on a neap day shifts by hours over 100 M. Close a strange coast in daylight: slow down or heave to rather than arrive before dawn. The one-triangle course gives the landfall time: the hours the arc needs to reach the coast.
7 · Fuel and range
Litres per mile, not per hour
How do I calculate a motor yacht's range from its fuel curve?
For each speed, hours = usable fuel ÷ litres per hour and range = hours × speed, usable being the tank less the reserve. Litres per mile, l/h ÷ speed, are lowest slow, below the hump.
Usable fuel is the tank less the reserve. For each speed on the curve, hours = usable ÷ l/h and range = hours × speed; l/h ÷ speed is litres per mile, lowest slow, below the hump, and among planing speeds well past it. Against a current, range over the ground is hours × (speed − current). A head sea costs 30% more per hour, or speed at the same revs.
Range from the curve: worked example
8 kn 40 l/h: 5.0 l/M · 12 kn 110 l/h: 9.2 l/M
16 kn 210 l/h: 13.1 l/M · 20 kn 220: 11.0 l/M
Usable 2000 l at 12 kn: 18.2 h, 218 M
On the plane, 20 kn: 9.1 h, 182 M
Foul current 2 kn at 12 kn: 18.2 × 10 = 182 M
Head sea: +30% per hour at the same speed
Fast first, then slow: worked example
200 M · usable 1800 l
15 kn 150 l/h · 8 kn 48 l/h x h fast: 150x + 48 × (200 − 15x) ÷ 8 = 1800 x = 10 h: 150 M at 15 kn, then 50 M at 8 kn
1500 + 300 = 1800 l: arrive on the reserve
Motor-sailing with the main drawing gives the same speed at lower revs, so the range grows. Dead to windward, motor-sailing 30° off and tacking makes good speed × cos(30° + leeway), often more than motoring straight into the sea.
Practice questions
Questions from this topic, as the app asks them. Open one to see the answer.
Log book
1800 fix · log 1204.0 · heading 210°T
2300 log 1241.5 · altered to 235°T
0600 log 1294.0
No stream, no leeway (running)
Your GNSS failed at 1800 halfway across Biscay. From the log book, where is the 0600 DR from the 1800 fix?
225°T, 87.9 M
045°T, 87.9 M
210°T, 90.0 M
223°T, 90.0 M
Show the answer
A. 225°T, 87.9 M
37.5 M on 210°T, then 52.5 M on 235°T, added as vectors: 87.9 M on 225°T. Not the mean course and the summed log.
All 100 questions of this topic are in the app: practice by topic with the reason after every answer, your mistakes brought back, and a timed mock exam. Free, no sign-up, in the browser or in Telegram.
