Mathematics 2023 · May/June

m a height of 9m above horizontal ground. The speed of the particle at the instant before hitting the ground is 12ms Find the work done against air resistance.

2Two particles AandB, of masses 3.2kg and 2.4kg respectively, lie on a smooth horizontal table. Amoves towardsBwith a speed ofvms and collides withB, which is moving towardsAwith a speed of 6ms . In the collision the two particles come to rest. (a)Find the value of v. [2] (b)Find the loss of kinetic energy of the system due to the collision.

15 N 33 N 30 N Coplanar forces of magnitudes 30N, 15N, 33N andPN act at a point in the directions shown in the diagram, where tan . The system is in equilibrium. (a)Show that 14.4 28.8 P+30 =1.

4An athlete of mass 84kg is running along a straight road. (a)Initially the road is horizontal and he runs at a constant speed of 3ms . The athlete produces a constant power of 60W. Find the resistive force which acts on the athlete. [1] (b)The athlete then runs up a 150m section of the road which is inclined at 0.8 ?to the horizontal. The speed of the athlete at the start of this section of road is3ms and he now produces a constant driving force of 24N. The total resistive force which acts on the athlete along this section of road has constant magnitude 13N.

35Å 0.6 kgPN A particle of mass 0.6kg is placed on a rough plane which is inclined at an angle of 35 ?to the horizontal. The particle is kept in equilibrium by a horizontal force of magnitude PN acting in a particle and plane is 0.4. Find the least possible value of P.

6A particle Pstarts at rest and moves in a straight line from a pointO. At timets after leavingO, the velocity of P,vms , is given byv=bt+ct 2, wherebandcare constants.Phas velocity 8ms whent=4 and has velocity 13.5ms whent=9. (a)Show that (b)Find the acceleration of Pwhent=1. [2] (c)Find the positive value of this instant.

(d)Find the speed of Pat the instant it returns toO.

0.25 kg 2 kg 1.2m0.8m Two particlesPandQ, of masses 2kg and 0.25kg respectively, are connected by a light inextensible Pis on an inclined plane at an angle of 30?to the horizontal. Particle Qhangs below the pulley. Three pointsA,BandClie on a line of greatest slope of the plane with AB=0.8m andBC=1.2m (see diagram). Particle Pis released from rest atAwith the string taut and slides down the plane. During the motion PfromAtoC,Qdoes not reach the pulley. The part of the plane fromAtoBis rough, with P. The part of the plane fromBtoCis smooth. (a) (i)Find the acceleration of PbetweenAandB.

PatC. [5] (b)Find the time taken for Pto travel fromAtoC.

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