CIE May 2021 9709 Mechanics Paper 43

CIE May 2021 9709 Mechanics Paper 43 (pdf)

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1. Particles P of mass 0.4 kg and Q of mass 0.5 kg are free to move on a smooth horizontal plane. P and Q are moving directly towards each other with speeds 2.5 m s⁻¹ and 1.5 m s⁻¹ respectively. After P and Q collide, the speed of Q is twice the speed of P.
   Find the two possible values of the speed of P after the collision
2. A cyclist is travelling along a straight horizontal road. She is working at a constant rate of 150 W. At an instant when her speed is 4 m s⁻¹, her acceleration is 0.25 m s⁻². The resistance to motion is 20 N.
   (a) Find the total mass of the cyclist and her bicycle
   The cyclist comes to a straight hill inclined at an angle 1 above the horizontal. She ascends the hill at constant speed 3 m s⁻¹. She continues to work at the same rate as before and the resistance force is unchanged.
   (b) Find the value of θ.
3. Four coplanar forces act at a point. The magnitudes of the forces are 20 N, 30 N, 40 N and F N. The directions of the forces are as shown in the diagram, where sin α ° = 0.28 and sin α ° = 0.6.
   Given that the forces are in equilibrium, find F and θ.
4. A particle is projected vertically upwards with speed μ m s⁻¹ from a point on horizontal ground. After 2 seconds, the height of the particle above the ground is 24 m.
   (a) Show that μ = 22.
   (b) The height of the particle above the ground is more than h m for a period of 3.6 s.
   Find h

5. A car of mass 1400 kg is towing a trailer of mass 500 kg down a straight hill inclined at an angle of 5Å to the horizontal. The car and trailer are connected by a light rigid tow-bar. At the top of the hill the speed of the car and trailer is 20 m s⁻¹ and at the bottom of the hill their speed is 30 m s⁻¹. (a) It is given that as the car and trailer descend the hill, the engine of the car does 150 000 J of work, and there are no resistance forces.
   Find the length of the hill.
   (b) It is given instead that there is a resistance force of 100 N on the trailer, the length of the hill is 200 m, and the acceleration of the car and trailer is constant.
   Find the tension in the tow-bar between the car and trailer.
6. A particle moves in a straight line and passes through the point A at time t = 0. The velocity of the particle at time ts after leaving A is vm s⁻¹, where
   v = 2t² − 5t + 3.
   (a) Find the times at which the particle is instantaneously at rest. Hence or otherwise find the minimum velocity of the particle.
   (b) Sketch the velocity-time graph for the first 3 seconds of motion.
   (c) Find the distance travelled between the two times when the particle is instantaneously at rest
7. A particle P of mass 0.3 kg rests on a rough plane inclined at an angle θ to the horizontal, where sin θ = 7/25. A horizontal force of magnitude 4 N, acting in the vertical plane containing a line of greatest slope of the plane, is applied to P (see diagram). The particle is on the point of sliding up the plane.
   (a) Show that the coefficient of friction between the particle and the plane is 3/4.
   
   The force acting horizontally is replaced by a force of magnitude 4 N acting up the plane parallel to a line of greatest slope.
   (b) Find the acceleration of P.
   (c) Starting with P at rest, the force of 4 N parallel to the plane acts for 3 seconds and is then removed. Find the total distance travelled until P comes to instantaneous rest.

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