The heart of dead reckoning

Time, speed, distance calculations are an integral part of any navigator’s skill set, forming the basis by which a navigator calculates a DR position, figures the estimated time of arrival (ETA), and derives speed from navigation plots. Most of us learned of the relationship between time, speed, and distance in fifth or sixth grade, when we were first introduced to the concept of word problems. On a boat, the elements of the equation remain the same. In this short discussion, we will look at a few ways to get at the simple yet important relationship between our boat speed, the distance we travel at that speed, and how long it takes (or will take) to get there.

The basic formula: distance = speed x time.

Vessel speed = 7 knots

Distance to travel = 9.2 nm

What is the ETA?

D = ST, 9.2 = 7 ? T

T = 9.2 ÷ 7

T = 1.31 hrs. (about one hour and 19 minutes)

A better version is: 60 (distance) = speed ? time. This version of the basic formula will yield an answer for time in minutes, when speed and distance are known (remember this formula by saying “sixty D street” as in “60D = ST”).

Same problem as above, but this time, let’s remember that we already know it will take us one hour to travel 7 nm, so all we have to calculate is how long it takes to travel 2.2 nm (9.2 – 7):

60(2.2) = 7 ? T

T = 60(2.2) ÷ 7

T = 19 minutes

total time = 1 hr. to travel 7 nm + 19 min. to travel

2.2 nm = 1 hr., 19 min. (1.3 hrs.)

The six-minute rule is a great tool for the cockpit navigator and is useful for a navigator looking for a fast way to lay down a series of short-interval DR positions. The six-minute rule is also a vital part of any radar collision avoidance plot. The rule states that the distance traveled in six minutes is equal to one-tenth of the distance traveled in an hour. Sounds almost too simple to bother with until faced with trying to calculate a distance traveled in five minutes (how many of us can quickly calculate one-twelfth of eight nautical miles?). Far simpler than mentally crunching through 60D = ST, the six-minute rule allows a quick movement of the decimal point one place to the left: for example, to find out how far we travel in six minutes at eight knots, just move the decimal point one place to the left, making 8.0 (the boat speed and the number of miles traveled in one hour) into 0.8.

The six-minute rule really comes into play in close-in harbor approach navigation, when precise position information and navigation decisions must be derived quickly. It is much easier to lay down a DR track at six minute intervals (or a multiple of six, since the rule can be used easily for 12-, 18-, or even three-minute intervals) than to struggle to find the calculator or the circular slide rule. With the six-minute rule the navigator can do all such calculations in his or her head.

Same problem as above, but this time using the six-minute rule: Again, we only need to find a T for a distance of 2.2 nm and then add 1 hour.

At seven knots, we travel 0.7 nm every six minutes, so 2.2 ÷0.7 = 3, or 3 six-minute increments, or 18 min. (plus a bit more).

Total time = 1 hr. to travel 7 nm + 18 min.

to travel 2.2 nm = 1 hr., 18 minutes

Scales: On most charts you’ll find a logarithmic speed scale (see figure 1). Although this scale takes some getting used to, a navigator who masters it can quickly find the answer to many time, speed, distance questions. The instructions accompany the scale, so a quick warm-up with them is helpful if the navigator forgets how to use the scale. This scale can also be cut from the chart and laminated for more convenient use (scales on a chart always seem to be under the fold). By the way, any scale, enlarged to any size, will work for any chart.As an example, let’s assume a distance traveled of 3.4 miles and an elapsed time of 17 minutes. Place one divider point on 3.4 and the other point on 17. Without changing the divider spread, place the right point on 60 and the left point will show speed, 12 knots in this example. This scale can be used to find any unknown of the time, speed, distance equation.

There is also a version of the logarithmic speed scale on the pre-printed Maneuvering Boards available from the DMA (figure 2). This version uses three scales (one each for time, speed, and distance) laid out one over the other. The navigator locates two known points on their appropriate scales and draws a line between them so that the line extends to arrive at the missing third point (for example, knowing time and distance points yields speed). Again, this is a pretty nifty scale, but unless it is readily accessible it is of little value to the navigator. Laminating a copy for easy use is recommended.

The best tool for making time, speed, distance calculations second nature is just plain practice. After all, the best time to learn an unfamiliar skill is before it is needed.