Chapter 2: Vectors and scalars
The chapter problem: the river ferry
A small ferry carries people across a river. The engine gives the ferry a speed of 4.0 m s⁻¹ through the water. The river is 120 m wide. The current flows at 3.0 m s⁻¹, at right angles to the crossing.
On the first crossing the ferry points straight across the river, towards the far bank. The river carries it sideways as it goes.
These are the facts to use all the way through the chapter.
- The river is 120 m wide.
- The ferry moves at 4.0 m s⁻¹ through the water.
- The current flows at 3.0 m s⁻¹, at right angles to the crossing.
- On the first crossing the ferry points straight across.
By the end of the chapter you will work out the ferry's velocity over the ground. You will find how long the crossing takes, and how far downstream the ferry lands. You will then find the heading that keeps the ferry on a straight line to the far bank, and decide which crossing the operator is likely to choose.
First contact: no working yet. Predict. With no current, the ferry would cross 120 m at 4.0 m s⁻¹ in 30 s. The current is at right angles to the crossing. Does the crossing take more than 30 s, less than 30 s, or about 30 s? Write down your answer and one sentence of reasoning.
If you are not sure where to start
Think about the two motions separately. The engine carries the ferry across the river. The current carries the ferry downstream. The two motions are at right angles, so the current does not add to or subtract from the across motion.
By the end of this chapter you will be able to
- tell a scalar quantity from a vector quantity;
- write a vector as a magnitude and a direction, and use positive and negative signs to give the direction of a vector along a line;
- resolve a vector into two perpendicular components;
- add two perpendicular vectors to find the resultant, using Pythagoras and the tangent;
- find a component when the vector and an angle are known, and find the angle when a component is known;
- use a scale drawing to add two vectors that are not at right angles.
Where it is used
Every route on a map is a sum of displacements. A pilot adds the aircraft's velocity through the air to the wind to find the path over the ground. A structural engineer adds the forces on a joint to check that they balance. A skier resolves weight into a component down the slope and a component into the snow. The same ideas return in every later chapter, where forces, velocities and fields are added and resolved.
Your turn to connect it
Think of a journey you make often. It could be a walk to school, a bus route or a cycle ride. Write down the total distance you travel, and the straight-line displacement from the start to the finish. Say why the two numbers are different.