Velocity is a vector; speed is its magnitude. To combine a velocity relative to moving water with the water velocity relative to land, add the two vectors. Example: a boat moves north at 4 m s-1 relative to water flowing east at 3 m s-1. Its velocity relative to land is 3 i+4 j, and its speed is 5 m s-1. For speed s at angle alpha anticlockwise from east, the components are scosalpha east and ssinalpha north. The signs identify the quadrant. For a bearing beta, measured clockwise from north, the velocity is ssinbeta, i+scosbeta, j. A bearing of 300 has negative east component and positive north component. If direction is given by 3 i+4 j and speed is 12, first divide by 5: the velocity is 365 i+ 485 j, not 36 i+48 j.

Velocity is a vector; speed is its magnitude. To combine a velocity relative to moving water with the water velocity relative to land, add the two vectors.

Example: a boat moves north at \(4\text{ m s}^{-1}\) relative to water flowing east at \(3\text{ m s}^{-1}\). Its velocity relative to land is \(3\mathbf i+4\mathbf j\), and its speed is \(5\text{ m s}^{-1}\).

For speed \(s\) at angle \(\alpha\) anticlockwise from east, the components are \(s\cos\alpha\) east and \(s\sin\alpha\) north. The signs identify the quadrant.

For a bearing \(\beta\), measured clockwise from north, the velocity is \(s\sin\beta\,\mathbf i+s\cos\beta\,\mathbf j\). A bearing of \(300^\circ\) has negative east component and positive north component.

If direction is given by \(3\mathbf i+4\mathbf j\) and speed is \(12\), first divide by \(5\): the velocity is \(\dfrac{36}{5}\mathbf i+\dfrac{48}{5}\mathbf j\), not \(36\mathbf i+48\mathbf j\).

Composing and resolving velocities

Composing and resolving velocities