Dead reckoning (DR and EP)

How to think it through

1You know where you started, your course, speed and time; you want your position.
2Split the leg into a north-south part (latitude changes) and an east-west part (longitude changes).
3Without tide you are at the DR. With tide, move from the DR by the drift in the direction of the set: that is the EP.

Worked example

Every step says why you do it, which formula you use, the working and the answer.

Estimated position (EP)

At 1000 you are at 49°55.0′N 002°46.1′W. You steer 075°T at 5.5 knots for 1 h 30 min. The tide sets 140° at 1.7 knots. Where are you?

Start: 49°55.0′N 002°46.1′W
True course and speed: 075°T · 5.5 kn
Time: 90 min
Tide (set, rate): 140° · 1.7 kn

How many miles do you run through the water?
Why: First you need the length of the leg: speed times time.
$D_{NM} = \frac{S _{kn} \times T _{min}}{60}$
Working: 5.5 kn × 90 min ÷ 60 = 8.3 NM
Answer: 8.3 NM
By how many minutes does the latitude change? (positive going north, negative going south)
Why: The north-south part of the leg tells you how far your latitude changes.
$d.lat_{^{'}} = D \times cos (course)$
Working: 8.3 × cos 075° = 2.1′
Answer: 2.1 ′
By how many minutes does the longitude change? (positive going east, negative going west)
Why: The east-west part is in miles; to turn it into minutes of longitude divide by the cosine of latitude, because meridians converge.
$d.long_{^{'}} = \frac{D \times sin ( course )}{cos φ _{mean}}$
Working: 8.3 × sin 075° = 8.0 miles; ÷ cos 49.9° = 12.4′
Answer: 12.4 ′
What is the DR position?
Why: Adding the two changes to the start gives where you would be with no tide.
$φ_{DR} = φ_{0} + d.lat λ_{DR} = λ_{0} + d.long$
Working: DR: 49°57.1′N 002°33.7′W
Answer: 49°57.1′N 002°33.7′W
How many miles does the tide carry you in this time (drift)?
Why: In the same time the tide carries you along: its rate times the time.
$Drift_{NM} = Rate_{kn} \times T_{h}$
Working: 1.7 kn × 90 ÷ 60 = 2.6 NM towards 140°
Answer: 2.6 NM
What is the estimated position (EP)?
Why: From the DR move by the drift, in the direction the tide sets: that is where you really are.
$EP = DR + drift towards the set$
Working: Total movement: 0.2′ of latitude and 9.6 miles of departure. EP: 49°55.2′N 002°31.2′W
Answer: 49°55.2′N 002°31.2′W

Now you try

With the examiner (even from a photo)