# A-level Physics (Advancing Physics)/Vectors/Worked Solutions

**1. Which of the following are vectors?**

**20 cm**is scalar, not a vector - it does not have a directional component.**9.81 ms**is a vector.^{-2}towards the centre of the earth**5 km south-east**is a vector.**500 ms**is a vector.^{-1}on a bearing of 285.3°

**2. A displacement vector** a **is the resultant vector of two other vectors, 5 m north and 10 m south-east. What does** a **equal, as a displacement and a bearing?**

Displacement (m) |
Bearing (°) |
Horizontal Displacement (m) |
Vertical Displacement (m)' |

5 | 000 | 0 | 5 |

10 | 135 | , so | , so |

So, for the resultant vector:

This gives us a right-angled triangle. So:

°, so the bearing equals 90° + 16.3° = 106.3°.

**3. If I travel at a velocity of 10 ms ^{-1} on a bearing of 030°, at what velocity am I travelling north and east?**

**4. An alternative method of writing vectors is in a column, as follows:**

,

**where x and y are the vertical and horizontal components of the vector respectively. Express |**a**| and the angle between** a **and** **in terms of x and y.**

By Pythagoras' theorem:

Let θ be the angle between **a** and .

This angle θ is known as the argument of **a**.

**5. A more accurate method of modelling the trajectory of a ball is to include air resistance as a constant force** F**. How would this be achieved?**

Once the arrow representing the acceleration due to gravity has been added on, add an horizontal arrow pushing against the motion of the ball. Since **F** = m**a**, the magnitude of this acceleration is **F** divided by **m**.

Note that this model is still not perfect. In fact, **F** is not constant - it depends on the horizontal component of the ball's velocity.