IGCSE Additional Math 13.4. Compose and resolve velocities Topic Practice

CAIE IGCSE Additional Math 13.4. Compose and resolve velocities question practice helps you revise this syllabus point with the course map in view. Use this page to focus on one topic, check the style of questions available, and connect each attempt back to the knowledge area it is testing.

EduNinja keeps Additional Math practice aligned to CAIE, so you can move from topic review into exam-style question bank work without losing the syllabus structure. Start with a small set, mark the weak steps, then return to nearby topic links when a definition, graph, calculation, or explanation needs repair.

13.4.1—Compose and resolve velocities

(a)

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

[ 2 ]

(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 v. Find the magnitude and bearing of v.

[ 5 ]

[Maximum number: 8]

In this question lengths are in centimetres and time, t, is in seconds.
A particle P is moving in a straight line with a speed of 26 in the direction of the vector $\binom{5}{-12}$ .

(a)

Find the velocity vector of P.

When t=0, P passes through a point A which has position vector $\binom{3}{6}$ .

[ 2 ]

(b)

Write down the position vector of P at time t.

At the same time that P passes through A, a particle Q passes through a point B.
The position vector of Q at time t is given by $\binom{8 t-5}{2-25 t}$ .
The distance between P and Q at time t is d.

[ 2 ]

(c)

Show that $d^{2}=m t^{2}+n t+r$ , where m, n and r are integers to be found.

[ 3 ]

(d)

Hence show that P and Q do not collide.

[ 1 ]

[Maximum number: 5]

In this question all distances are in km.
A ship P sails from a point A, which has position vector $\binom{0}{0}$ , with a speed of $52 \mathrm{kmh}^{-1}$ in the direction of $\binom{-5}{12}$ .

(a)

Find the velocity vector of the ship.

[ 1 ]

(b)

Write down the position vector of P at a time t hours after leaving A.

At the same time that ship P sails from A, a ship Q sails from a point B, which has position vector $\binom{12}{8}$ , with velocity vector $\binom{-25}{45} \mathrm{kmh}^{-1}$ .

[ 1 ]

(c)

Write down the position vector of Q at a time t hours after leaving B.

[ 1 ]

(d)

Find the value of t when P and Q are first 2 km apart.

[ 2 ]

[Maximum number: 5]

In this question, all lengths are in metres.

(a)

A particle P has position vector $\binom{2+12 t}{5-5 t}$ at a time t seconds, $t \geqslant 0$ .

[ 5 ]

(i)

Write down the initial position vector of P .

[ 1 ]

(ii)

Find the speed of P.

[ 2 ]

(iii)

Determine whether P passes through the point with position vector $\binom{158}{-48}$ .

[ 2 ]

[Maximum number: 9]

In this question, time is in seconds.

(a)

At time t=0, particle P starts from the point with position vector -30 j.
P travels with speed $58 \mathrm{~ms}^{-1}$ in the direction 20 i+21 j.
Find the position vector of P at time t.

[ 3 ]

(b)

Also at time t=0, particle Q starts from the point with position vector -10 i+18 j.
Q travels with speed $75 \mathrm{~ms}^{-1}$ at an angle $\alpha$ above the positive x-axis, where $\tan \alpha=\frac{7}{24}$ .
Find the position vector of Q at time t.

[ 4 ]

(c)

Determine whether P and Q collide.

T
[2]

[ 2 ]

[Maximum number: 9]

In this question i is a unit vector in the positive x-direction and j is a unit vector in the positive y-direction. Time is in seconds and distances are in metres.

The diagram shows the initial positions and velocities of two particles, A and B, that move in the x-y plane.

Particle A starts from the origin O at time t=0. It moves with constant speed $10 \mathrm{~ms}^{-1}$ in the direction $60^{\circ}$ above the x-axis.

(a)

Find the exact values of the components of the velocity of particle A in the x-direction and the y-direction.

[ 2 ]

(b)

Find, in terms of t, the position vector of particle A at time t.

Particle B starts from the point $(2 \sqrt{3}, 9)$ at time t=0. It moves with constant speed $\frac{5}{3} \mathrm{~ms}^{-1}$ parallel to the positive x-axis.

[ 1 ]

(c)

Find, in terms of t, the position vector of particle B at time t.

[ 2 ]

(d)

Hence show that the particles collide.

[ 4 ]

[Maximum number: 7]

In this question, the x - and y-directions are east and north respectively. The units are metres and seconds. Boat A starts from the origin O and moves with constant speed $5 \sqrt{3} \mathrm{~ms}^{-1}$ on a bearing of $030^{\circ}$ . After 100 seconds boat B starts from point P, which has position vector $\binom{0}{1000}$ .
Boat B moves with constant speed $10 \mathrm{~ms}^{-1}$ on a bearing of $060^{\circ}$ .

(a)

Find the velocity of each boat in vector form.

[ 2 ]

(b)

Show that the two boats will collide.

[ 5 ]

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Question 6

Question 6(a)

Question 6(b)

Question 6

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Question 6(c)

Question 6(d)

Question 8

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Question 8(c)

Question 8(f)

Question 9

Question 9(a)

Question 9(a)(i)

Question 9(a)(ii)

Question 9(a)(iii)

Question 10

Question 10(a)

Question 10(b)

Question 10(c)

Question 11

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Question 11(b)

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Question 12

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