Adding velocities of current and ship as vectors

Question

I'm just struggling with my trigonometry homework. If someone could explain what the question is asking, it would be much appreciated.

A ship sails due north (relative to the current) at a speed of 20 knots. The water itself is moving northeast at a speed of 10 knots. What is the ship's velocity vector?

Answer 1

The question is asking for the ship's speed and direction (these are the two things that every vector has, remember) relative to the Earth.

The ship itself is sailing due north at 20 knots, but you also need to account for how quickly and in what direction the current is pushing the ship off course.

The ship's velocity vector should point somewhere between due north and due northeast with a magnitude (speed) greater than both 10 and 20 knots. Drawing a picture usually helps with this kind of question.

Answer 2

Velocity of ship wrt an observer on the ground is the vector sum of $v_{sc}$ and $v_{cg}$. Note this is a vector addition, not an algebraic equation.

Method 1: Geometric Approach

1. To find the magnitude of ship velocity, use cosine law.

2. To find the direction of ship velocity, use sine law.

Method 2: Define Cartesian coordinates

1. Find the components of $v_{sc}$ and $v_{cg}$

2. Find $v_{sg}$ by adding the components of $v_{sc}$ and $v_{cg}$ algebraiclly: $v_{sg,x} = v_{sc,x} + v_{cg,x}\dots$