CAIE Mathematics (9709)

A particle P moves in a straight line. At time t s after leaving a point O on the line, P has velocitymsv , where vt t44636 =- -.

Find the set of values of t for which the acceleration of the particle is positive.

Find the two values of t at which P returns to O. [3] 25°40° 2 N 16 N 10 NP N Four coplanar forces of magnitude P N, 10 N, 16 N and 2 N act at a point in the directions shown in the diagram. It is given that the forces are in equilibrium. Find the values of θ and P.

A car has mass 1400 kg. When the speed of the car is msv the magnitude of the resistance to motion is Nkv where k is a constant.

The car moves at a constant speed of ms24 up a hill inclined at an angle of a to the horizontal where .sin0 12a= . At this speed the magnitude of the resistance to motion is 480 N.

Find the value of k.

Find the power of the car's engine.

The car now moves at a constant speed on a straight level road. Given that its engine is working at kW54, find this speed. [3] 35° 0.8 kg28°T N A particle of mass 0.8 kg lies on a rough plane which is inclined at an angle of 28° to the horizontal. The particle is kept in equilibrium by a force of magnitude T N. This force acts at an angle of 35° above a line of greatest slope of the plane (see diagram). The coefficient of friction between the particle and the plane is 0.2 . Find the least and greatest possible values of T.

Three particles A, B and C of masses 5 kg, 1 kg and 2 kg respectively lie at rest in that order on a straight smooth horizontal track XYZ. Initially A is at X, B is at Y and C is at Z. Particle A is projected towards B with a speed of ms6 and at the same instant C is projected towards B with a speed of msv . In the subsequent motion, A collides and coalesces with B to form particle D. Particle D then collides and coalesces with C to form particle E and E moves towards Z.

Show that after the second collision the speed of E is ms 15 1- -

The total loss of kinetic energy of the system due to the two collisions is 63 J. Use the result from

to show that v3=.

It is given that the distance XY is 36 m and the distance YZ is 98 m.

Find the time between the two collisions.

Find the time between the instant that A is projected from X and the instant that E reaches Z. [1] 30° 2.5 kg 0.5 kg 2 m Two particles P and Q of masses 2.5 kg and 0.5 kg respectively are connected by a light inextensible string that passes over a small smooth pulley fixed at the top of a plane inclined at an angle of 30° to the horizontal. Particle P is on the plane and Q hangs below the pulley such that the level of Q is 2 m below the level of P (see diagram). Particle P is released from rest with the string taut and slides down the plane. The plane is rough with coefficient of friction 0.2 between the plane and P.

Find the acceleration of P.

Use an energy method to find the speed of the particles at the instant when they are at the same vertical height. [5] Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown. Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge Assessment International Education Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cambridgeinternational.org after the live examination series. Local Examinations Syndicate (UCLES), which is a department of the University of Cambridge.