Pearson Edexcel GCE Advanced Level In Mathematics (9MA0) Paper 32: Mechanics Plus Mark Scheme- From page 21.

Pearson Edexcel Level 3 GCE Monday 19 October 2020 2 *P66789A0220* 1. A rough plane is inclined to the horizontal at an angle α, where tanα = 3 4 A brick P of mass m is placed on the plane. The coefficient of fric ...

Pearson Edexcel Level 3 GCE Monday 19 October 2020 2 *P66789A0220* 1. A rough plane is inclined to the horizontal at an angle α, where tanα = 3 4 A brick P of mass m is placed on the plane. The coefficient of friction between P and the plane is μ Brick P is in equilibrium and on the point of sliding down the plane. Brick P is modelled as a particle. Using the model, (a) find, in terms of m and g, the magnitude of the normal reaction of the plane on brick P (2) (b) show that μ = 3 4 (4) For parts (c) and (d), you are not required to do any further calculations. Brick P is now removed from the plane and a much heavier brick Q is placed on the plane. The coefficient of friction between Q and the plane is also 3 4 (c) Explain briefly why brick Q will remain at rest on the plane. (1) Brick Q is now projected with speed 0.5ms-1 down a line of greatest slope of the plane. Brick Q is modelled as a particle. Using the model, (d) describe the motion of brick Q, giving a reason for your answer. (2) ( 2. A particle P moves with acceleration (4i - 5j)ms-2 At time t = 0, P is moving with velocity (-2i + 2j)m s-1 (a) Find the velocity of P at time t = 2 seconds. (2) At time t = 0, P passes through the origin O. At time t = T seconds, where T > 0, the particle P passes through the point A. The position vector of A is (λi - 4.5j)m relative to O, where λ is a constant. (b) Find the value of T. (4) (c) Hence find the value of λ (2) 3. (i) At time t seconds, where t  0 , a particle P moves so that its acceleration ams-2 is given by a = (1 - 4t) i + (3 - t2) j At the instant when t = 0, the velocity of P is 36i ms-1 (a) Find the velocity of P when t = 4 (3) (b) Find the value of t at the instant when P is moving in a direction perpendicular to i (3) (ii) At time t seconds, where t  0 , a particle Q moves so that its position vector r metres, relative to a fixed origin O, is given by r = (t2 - t) i + 3tj Find the value of t at the instant when the speed of Q is 5 ms-1 (6) 4. A C α 6a 4 a B Figure 1 A ladder AB has mass M and length 6a. The end A of the ladder is on rough horizontal ground. The ladder rests against a fixed smooth horizontal rail at the point C. The point C is at a vertical height 4a above the ground. The vertical plane containing AB is perpendicular to the rail. The ladder is inclined to the horizontal at an angle α, where sinα = 4 5 , as shown in Figure 1. The coefficient of friction between the ladder and the ground is μ. The ladder rests in limiting equilibrium. The ladder is modelled as a uniform rod. Using the model, (a) show that the magnitude of the force exerted on the ladder by the rail at C is 9 25 Mg (3) (b) Hence, or otherwise, find the value of μ. (7) ( 5. N 45° O m 25 A Ums-1 100m Figure 2 A small ball is projected with speed Ums-1 from a point O at the top of a vertical cliff. The point O is 25 m vertically above the point N which is on horizontal ground. The ball is projected at an angle of 45° above the horizontal. The ball hits the ground at a point A, where AN = 100m, as shown in Figure 2. The motion of the ball is modelled as that of a particle moving freely under gravity. Using this initial model, (a) show that U = 28 (6) (b) find the greatest height of the ball above the horizontal ground NA. (3) In a refinement to the model of the motion of the ball from O to A, the effect of air resistance is included. This refined model is used to find a new value of U. (c) How would this new value of U compare with 28, the value given in part (a)? (1) (d) State one further refinement to the model that would make the model more realistic. (1)