Compose & Resolve Velocities (Cambridge (CIE) O Level Additional Maths): Revision Note

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 $\left(\begin{array}{c}3\\ 7\end{array}\right)\mathrm{m}{\mathrm{s}}^{-1}$ (or $\left(3\mathbf{i}+7\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-1}$) would represent a particle or object moving

    • at $3\mathrm{m}{\mathrm{s}}^{-1}$ (metres per second) in the positive $x$ direction

    • and $7\mathrm{m}{\mathrm{s}}^{-1}$ 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 $\left(\begin{array}{c}6\\ 8\end{array}\right)\mathrm{m}{\mathrm{s}}^{-1}$ is $\sqrt{6^2+8^2}=10\mathrm{m}{\mathrm{s}}^{-1}$

    • An object travelling with velocity $\left(\begin{array}{c}-6\\ -8\end{array}\right)\mathrm{m}{\mathrm{s}}^{-1}$ would have the same speed, $10\mathrm{m}{\mathrm{s}}^{-1}$ 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, $\left(x\mathbf{i}+y\mathbf{j}\right)\mathrm{m}$ (metres, from the origin) at time $t$ seconds after its motion has started

  • This position vector is often called $\mathbf{r}$

• If the particle moves with (constant) velocity $\mathbf{v}\mathrm{m}{\mathrm{s}}^{-1}$, after $t$ seconds its position vector will be

  • $\mathbf{r}={\mathbf{r}}_{\mathbf{0}}+\mathbf{v}t$ where ${\mathbf{r}}_{\mathbf{0}}$ 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 $\left(2,-3\right)$ and moves with (constant) velocity $\left(4\mathbf{i}+7\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-1}$ after 10 seconds its position will be

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

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 (${\mathbf{r}}_{\mathbf{0}}$) is not necessarily the origin

    • $\mathbf{r}={\mathbf{r}}_{\mathbf{0}}+\mathbf{v}t$

• 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