CAIE IGCSE Additional Math 13.4 compose and resolve velocities

Use this velocity-vectors page to model moving particles or ships with position vectors, time parameters and distance equations.

[Maximum number: 7]

(a)

In this question, $\mathbf{i}$ is a unit vector due east and $\mathbf{j}$ is a unit vector due north.
A cyclist rides at a speed of $4\,\mathrm{m\,s^{-1}}$ on a bearing of $015^\circ$ . Write the velocity vector of the cyclist in the form $x\mathbf{i}+y\mathbf{j}$ , where x and y are constants.

[ 2 ]

$4\cos75^\circ\,\mathbf{i}+4\sin75^\circ\,\mathbf{j}$
or $4\sin15^\circ\,\mathbf{i}+4\cos15^\circ\,\mathbf{j}$ oe, isw
B2
B1 for $x=4\cos75^\circ$ or $x=4\sin15^\circ$ oe, soi, or $y=4\sin75^\circ$ or $y=4\cos15^\circ$ oe, soi.
SC1 for a correct expression with missing brackets such as $\sqrt6-\sqrt2\,\mathbf{i}+\sqrt6+\sqrt2\,\mathbf{j}$ , or for (1.04,3.86) oe.

(b)

A vector of magnitude 6 on a bearing of $300^\circ$ is added to a vector of magnitude 2 on a bearing of $230^\circ$ to give a vector $\mathbf{v}$ . Find the magnitude and bearing of $\mathbf{v}$ .

[ 5 ]

$(-6\cos30^\circ-2\cos40^\circ)\mathbf{i}+(6\sin30^\circ-2\sin40^\circ)\mathbf{j}$ oe, soi
B1
$r^2=(-6.7282\ldots)^2+(1.7144\ldots)^2$
M1
r=6.94
or $6.9432329\ldots$ rot to 4 or more sf
A1
$\alpha=\tan^{-1}\left(\frac{6.7282\ldots}{1.7144\ldots}\right)$ or awrt 75.7
M1
Bearing $284^\circ$ or $284.2^\circ$ rot to 4 or more sf
A1

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13.4. Compose and resolve velocities question 1

Question (a)

Question (b)