Compose & Resolve Velocities | Cambridge O Level Additional Maths Revision Notes 2025

Modelling Velocities with Vectors

How are velocities modelled by vectors?

Although introduced as being about paths and distances between points, a vector can also represent a velocity
- For example the velocity vector $(\begin{matrix} 3 \\ 7 \end{matrix}) m s^{- 1}$ (or $(3 i + 7 j) m s^{- 1}$ ) would represent a particle or object moving
  - at $3 m s^{- 1}$ (metres per second) in the positive $x$ direction
  - and $7 m s^{- 1}$ in the positive $y$ direction
Speed is different to velocity
- Speed is a scalar quantity, velocity is a vector quantity
- Velocity has direction, as well as magnitude
- Speed is the magnitude of velocity
  - Therefore, speed can be found from velocity by finding its magnitude - i.e. using Pythagoras' theorem
    e.g. The speed of a particle travelling with velocity $(\begin{matrix} 6 \\ 8 \end{matrix}) m s^{- 1}$ is $\sqrt{6^{2} + 8^{2}} = 10 m s^{- 1}$
  - An object travelling with velocity $(\begin{matrix} - 6 \\ - 8 \end{matrix}) m s^{- 1}$ would have the same speed, $10 m s^{- 1}$ but be moving in the opposite direction

How do I find the position of a particle from a velocity vector?

In general, problems refer to a particle's position vector, $(x i + y j) m$ (metres, from the origin) at time $t$ seconds after its motion has started
- This position vector is often called $r$
If the particle moves with (constant) velocity $v m s^{- 1}$ , after $t$ seconds its position vector will be
- $r = r_{0} + v t$
  where $r_{0}$ is the initial position of the particle (i.e. its position at the start of the motion)
For example if a particle starts at the point with coordinates $(2, - 3)$ and moves with (constant) velocity $(4 i + 7 j) m s^{- 1}$ after 10 seconds its position will be

$\begin{array}{rcl} r & = & (2 i - 3 j) + 10 (4 i + 7 j) \\ r & = & ((2 + 40) i + (- 3 + 70) j) \\ r & = & (42 i + 67 j) m \end{array}$

How do I solve problems involving velocities and vectors?

Solving problems involving velocity may involve using a variety of the skills covered in the vectors section
- A resultant velocity may be comprised of two (or more) velocities
  - e.g. the velocity of a javelin will be influenced by the athlete's ability and the wind speed/direction
  - a resultant velocity is found by adding velocity vectors
Problems may be phrased to distinguish the difference between speed and velocity
- Speed is the magnitude of velocity (use Pythagoras' theorem)
Problems may use position vectors
- The initial position ( $r_{0}$ ) is not necessarily the origin
  - $r = r_{0} + v t$
There could be two particles to deal with in a problem
- be clear about which particle has which velocity, position, etc
  - If two particles collide at time $t$ seconds, then (at time $t$ ) their position vectors will be equal
  - Two vectors are equal if their components are equal

Examiner Tip

Vector diagrams drawn previously to show paths and distances can still be used to visualise velocities
- So use any given diagram, and if there isn't one, draw one!
Read questions carefully - a common mistake is to give a final answer as a position vector when the question has asked for coordinates, or vice versa
- e.g. A particle with position vector $(3 i - 2 j)$ has coordinates $(3, - 2)$

Worked example

A boat leaves a harbour (the origin) and sails with a (constant) resultant velocity comprising of the velocity produced by the boat's engine, $(i + 2 j) m s^{- 1}$ and the velocity produced by the water current, $(1.5 i + 2 j) m s^{- 1}$ .

Find the resultant velocity of the boat.

The resultant velocity will be the sum of the two velocities.

$v = (i + 2 j) + (1.5 i + 2 j) v = (2.5 i + 4 j)$

The resultant velocity of the boat is $(2.5 i + 4 j) m s^{- 1}$ .

A second boat has position vector $(27 i + 6 j) m$ at the same time as when the first boat leaves the harbour.
The second boat sails with (constant) resultant velocity $(- 2 i + 3 j) m s^{- 1}$ .

Without intervention, the two boats will collide at time

t

seconds.

Find the value of

t

ii)

Find the coordinates of the point at which the boats will collide.

The position of a particle at time

t

r = r_{0} + t v

where

r_{0}

is the initial position and

v

is its (resultant) velocity.

The two boats will be in the same position when they collide.

∴ (0 i + 0 j) + t (2.5 i + 4 j) = (27 i + 6 j) + t (- 2 i + 3 j)

Equating

i

(or

j

) components gives an equation in

t

\begin{array}{rcl} 2.5 t & = & 27 - 2 t \\ 4.5 t & = & 27 \\ t & = & 6 \end{array}

\begin{array}{rcl} 4 t & = & 6 + 3 t \\ t & = & 6 \end{array}

The boats collide at $t = 6$ seconds.

ii)

Find the position of either boat at

t = 6

seconds.

\begin{array}{rcl} r & = & (0 i + 0 j) + 6 (2.5 i + 4 j) \\ r & = & (15 i + 24 j) \end{array}

The question asks for coordinates (rather than a position vector).

The boats will collide at the point $(15, 24)$ .

Compose & Resolve Velocities (Cambridge O Level Additional Maths)

Revision Note

Modelling Velocities with Vectors

How are velocities modelled by vectors?

How do I find the position of a particle from a velocity vector?

How do I solve problems involving velocities and vectors?

Examiner Tip

Worked example

You've read 0 of your 10 free revision notes

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Paul