Advanced level 2025 South West Regional Mock mathematics mechanics 3

Advanced level 2025 South West Regional Mock mathematics mechanics 3

Advanced level 2025 South West Regional Mock mathematics mechanics 3

Here is the extracted data from the image:

1. The position vector of a particle P of mass 3 kg at time $t$ seconds is r” />=(3+sin2t)i^+(cos2t)j^​ m. a) Find the value of $t$ when P crosses the x-axis for the first time. b) Find the velocity and acceleration of P at time $t$. c) Show that the speed of the particle at any time is constant. d) Find the magnitude of the force acting on the particle when t=2π​.

2. (i) A system is made up of the forces: F” />1​=(3i^+2j^​+2k^) N acting at the points with position vectors r” />1​=(2i^−2j^​+3k^) m and F” />2​=(3i^−k^) N acting at r” />2​=(6i^+2j^​−k^) m respectively. Show that the system is in equilibrium. (ii) A non uniform ladder AB of length 8a and weight W is such that its weight acts at the point G on the ladder, where AG=47a​. The ladder rests with the end A on a rough horizontal ground and the end B on a smooth vertical wall, making an angle of 45∘ with the horizontal. The coefficient of friction between the ladder and the ground is $\mu$. A man of weight 4W stands at a point on the ladder, a distance $3a$ from the foot. His son, of weight 2W also starts climbing slowly. Show that the ladder begins to slip when the distance on the ladder between the man and his son is 10μ11a​.

3. (i) A smooth sphere of mass 2 kg is moving with velocity (2i^−3j^​) ms$^{-1}$ when it is struck by a smooth surface. As a result of this impact, the sphere receives an impulse of (4i^+8j^​) kgms$^{-1}$ and moves with velocity (ai^+bj^​) ms$^{-1}$. Find the values of the constants $a$ and $b$. (ii) A smooth sphere A of mass $m$ moving on a smooth horizontal table with velocity $2u$ when it collides directly with another smooth sphere B of mass $3m$ moving in the same direction as A on the same table with velocity $u$. After the collision, B moves with velocity 35u​ in the same direction, where $k$ is a constant. Find, in terms of $u$ and $k$: a) the velocity of A immediately after the collision. b) the coefficient of restitution between the spheres. c) Hence, or otherwise, deduce that 0≤k≤35​.

4. The engine of a motor bike of mass 300 kg is working at a constant rate of 54kW, against a total non-gravitational resistance of magnitude (a+bv2) N, where $v$ is its speed in metres per second and $a$ and $b$ are constants. The maximum speed on the horizontal stretch of road is 60 ms$^{-1}$ while its maximum speed on the stretch of road inclined at sin−1(101​) to the horizontal is 50 ms$^{-1}$. a) Find the values of the constants $a$ and $b$. b) Determine the acceleration of the motorbike on a straight level road at the instant when its speed is 30 ms$^{-1}$.

5. A particle of mass 0.5 kg is projected with speed $u$ from a point O, on a level horizontal ground, 40 m away from the foot of a vertical wall. The particle just clears the wall and another parallel wall before striking the ground again at the point A. The distance between the walls is 40 m and each wall has a height of 20 m. The plane in which the particle travels is perpendicular to the walls. a) Find the angle of projection of the particle. b) Show that u=252​ ms$^{-1}$. c) Calculate the kinetic energy of the particle just when it reaches maximum height.