Advanced level 2026 south west regional mock mathematics with mechanics 3

Here is the extracted text from the fifth image, which appears to be a Physics or Mechanics examination paper:

1. Two identical particles $P$ and $Q$ each of mass 2 kg move in the $x, y$-coordinate plane such that at time $t$, their velocities are $\mathbf{v}_1$ and $\mathbf{v}_2$ respectively, where

   $$\mathbf{v}_1 = [(2t)\mathbf{i} – 3t^2\mathbf{j}] \text{ ms}^{-1} \text{ and } \mathbf{v}_2 = [t^3\mathbf{i} + (2t – 3)\mathbf{j}] \text{ ms}^{-1}$$

   Find,

   (a) the velocity of $P$ relative to $Q$ when $t = 2$, (2 marks)

   (b) the non-zero values of $t$ when the direction of motion of $P$ is perpendicular to $Q$, (2 marks)

   (c) the force acting on $P$ when $t = 2$, (4 marks)

   Initially, both particles are at the origin, find

   (d) the distance between the particles when $t = 2$. (5 marks)

2. (i) A uniform rod $AB$ of length $2l$ and mass $m$ is smoothly hinged at $A$ on a vertical wall and kept in equilibrium in a horizontal position by a light inextensible string $BC$, where $C$ is vertically above $A$. The string makes an angle $\theta$ with the rod, where $\tan \theta = \frac{1}{2}$, show that

   (a) the tension in the string is of magnitude $\frac{\sqrt{5}}{2}mg$. (4 marks)

   (b) the magnitude of the reaction at the hinge is same as the tension in the string. (4 marks)

   (ii) A particle of mass 300 g is in circular motion round a circular path of radius 80 cm. Given that the particle completes one revolution in 1.2 s, find the centripetal force acting on the particle. (5 marks)

3. Two particles $A$ and $B$ of respective masses $\frac{11}{2}m$ and $\frac{9}{2}m$ are connected by a light inextensible string passing over a smooth fixed pulley. The system is released from rest with both particles vertical and the string taut, when $A$ is at a distance $d$ from the horizontal ground.

   Find

   (a) the acceleration of the system and the tension in the string. (5 marks)

   (b) the magnitude of the reaction force at the pulley. (2 marks)

   If a time $t$ elapses before $A$ hits the ground,

   (c) show that $20d = gt^2$. (3 marks)

   (d) find the speed with which $A$ strikes the ground. (3 marks)

Would you like me to help you set up the free-body diagram for the rod in question 2(i) or solve the pulley system in question 3?