Advanced level 2025 North West Regional Mock mathematics mechanics 3

Advanced level 2025 North West Regional Mock mathematics mechanics 3

Advanced level 2025 North West Regional Mock mathematics mechanics 3

Here is the extracted data from the image:

Problem 1:

• The position vector of a particle A at time $t$ seconds with respect to the Origin O is given by r” />=(2t−1)i^+t2j^​ m.
  • a) Find the position vector of A at time $t=2$.
  • b) Find the velocity of A at time $t$.
• The velocity of a second particle B relative to A at time $t$ is given by (2t+2)j^​ ms$^{-1}$. Initially B is at the point $(1,-1)$ m.
  • c) Find the velocity of B at time $t$. Hence show that its initial speed is 22​ ms$^{-1}$.
  • d) Find the position vector of B at time $t$.

Problem 2:

• A sphere of mass $m$, is moving on a smooth horizontal surface with speed $3u$, when it collides with a vertical wall. When the sphere reaches a point P, it receives an impulse in the opposite direction of its motion, of magnitude $mu$.
  • a) Deduce that the impulse slows down the sphere but does not change the direction of motion.
  • b) Calculate the change in kinetic energy experienced by the sphere.
• Given that the point P is at distance $d$ from the wall, and the coefficient of restitution between the sphere and the wall is $e$. Find:
  • c) the time it takes the sphere to travel from P to strike the wall.
  • d) the total time taken to travel from P to strike the wall and return to P.

Problem 3:

• A particle is projected from a point O on the horizontal through O, and allowed to move under gravity. Given that after 4 seconds, the particle passes through the point (32, 4), find:
  • a) the speed of projection.
  • b) the initial angle of projection and the magnitude of the velocity when $t=1$. Hence or otherwise, deduce whether the particle is moving upward or downward.
  • c) for how long the particle will be at a height above 4m.
  • (Take $g=10$ ms$^{-2}$)

Problem 4:

• i. A particle moves in a circular path with constant angular acceleration $\alpha$ rad s$^{-2}$. Given that its initial angular velocity is $2\pi$ rad s$^{-1}$ and that it turns through an angle of $6\pi$ rads in 5 s. Find the value of $\alpha$.
• ii. A car of mass 800 kg moves on a level road. The engine exerts a constant power of 40 kW. The resistance to motion is constant and equal to 500 N.
  • a) Find the initial acceleration of the car.
  • b) Find the maximum speed of the car.
  • c) Find the time taken by the car to reach a speed of 20 ms$^{-1}$ from rest.