Heading Problems

Heading problems are navigation problems that arise when pilots and boaters must correct for winds or currents. A simple example would be when someone wants to cross a river to get to the opposite side as shown in the diagram below.

If the person were to try to travel straight across, due North in this example, the river current would carry them west of where he or she wanted to be as shown below.

So to travel directly across, the person must actually head upriver shown in the following diagram.


In each case we are dealing with relative velocities. A relative velocity is a velocity measured with respect to a particular point or surface. That point or surface can be moving. A speedometer and compass on the boat would give the velocity of the boat with respect to the water. If a person on the shore measured the velocity of the boat by using a radargun such as the police use, that velocity would be with respect to the shore. These two velocities are related. The relationship is expressed in the formula:

and shown graphically by the diagram

Note that Vshore is the vector addition of the velocities of the boat and the river. In the diagram,the tail of Vriver is added to the nose of Vboat.

Usually in heading problems, we are given at least one distance. Since we deal with constant velocity (zero acceleration), the vector straight-line distance is the same as the displacement and is given by D = Vt where t is the travel time. The travel time is the same for all three velocities – every observer on the shore, in the boat, in the water will agree on how long the boat has been moving. As a result it is usually more useful to deal with the vector distances involved as shown in the diagram below

In Physics 1100m heading problems come in two types:

  1. Find the angle q upriver that a person must head to get directly across the river.

    The diagram below illustrates the solution to this case when the river flows east to west (←).

    Note that Dshore would be the width of the river in this case.

  2. Find where the point lands if the boater aims straight across. Alternately — what is the quickest time to cross the river.

    The diagram below illustrates the solution to this case when the river flows east to west (←).

    Note that Dboat would be the width of the river in this case.

Questions? mike.coombes@kwantlen.ca

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