CAIE Additional Mathematics Velocity Vectors

[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 $\binom{5}{-12}$ .

All through this question allow correctly written vectors in $\mathbf{i},\mathbf{j}$ form;
incorrectly written column vectors do not earn full marks unless the correct form is recovered.

(a)

Find the velocity vector of P.

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

[ 2 ]

$\binom{10}{-24}$ oe, isw
B1 for the magnitude of the direction vector is 13 soi, or M1 for $\sqrt{5^2+(-12)^2}$ oe.
A1

(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 ]

$\binom{3}{6}+t\binom{10}{-24}$ oe, isw
M1 FT for $\binom{3}{6}+t(\text{their }\binom{10}{-24})$ .
A1

(c)

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

[ 3 ]

$\binom{3+10t}{6-24t}-\binom{8t-5}{2-25t}=\binom{8+2t}{4+t}$ oe, soi
Finds $d^2$ or d for the difference of their position vectors.
M1 FT.
$d^2=5t^2+40t+80$ mark final answer

(d)

Hence show that P and Q do not collide.

[ 1 ]

Correct argument including d, e.g. when d=0, t=-4, but $t\geqslant0$ , so no collision;
or d>0 when $t\geqslant0$ , so $d\ne0$ .
B1 dep on part (c) being correct.

CAIE IGCSE Additional Math 13.4 Compose and Resolve Velocities Questions

Exam points

CAIE IGCSE Additional Math 13.4 Compose and Resolve Velocities Questions question 1

Question (a)

Question (b)

Question (c)

Question (d)