2024 IGCSE Additional Mathematics October/November 0606/22 Q11

In this question \(\mathbf i\) is a unit vector in the positive \(x\)-direction and \(\mathbf j\) 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 \(10\text{ m s}^{-1}\) in the direction \(60^\circ\) above the \(x\)-axis.

1. Find the exact values of the components of the velocity of particle \(A\) in the \(x\)-direction and the \(y\)-direction.[2]
2. Find, in terms of \(t\), the position vector of particle \(A\) at time \(t\).[1]

Particle \(B\) starts from the point \((2\sqrt3,9)\) at time \(t=0\). It moves with constant speed \(\dfrac53\text{ m s}^{-1}\) parallel to the positive \(x\)-axis.

3. Find, in terms of \(t\), the position vector of particle \(B\) at time \(t\).[2]
4. Hence show that the particles collide.[4]