Edexcel A Level Maths M2 · Mechanics 2 January 2008 question paper 6678

A parcel of mass 2.5 kg is moving in a straight line on a smooth horizontal floor. Initially the parcel is moving with speed 8 m s^–1. The parcel is brought to rest in a distance of 20 m by a constant horizontal force of magnitude R newtons. Modelling the parcel as a particle, find

the kinetic energy lost by the parcel in coming to rest,

the value of R.

At time t seconds (t 0), a particle P has position vector p metres, with respect to a fixed origin O, where

Find

the velocity of P at time t seconds,

the value of t when P is moving parallel to the vector i.

When t = 1, the particle P receives an impulse of (2 i – 6 j) N s. Given that the mass of P is 0.5 kg,

find the velocity of P immediately after the impulse.

A car of mass 1000 kg is moving at a constant speed of 16 m s^–1 up a straight road inclined at an angle θ to the horizontal. The rate of working of the engine of the car is 20 kW and the resistance to motion from non-gravitational forces is modelled as a constant force of magnitude 550 N.

When the car is travelling up the road at 16 m s^–1, the engine is switched off. The car comes to rest, without braking, having moved a distance y metres from the point where the engine was switched off. The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude 550 N.

Find the value of y.

A set square S is made by removing a circle of centre O and radius 3 cm from a triangular piece of wood. The piece of wood is modelled as a uniform triangular lamina ABC, with ABC = 90°, AB = 12 cm and BC = 21 cm. The point O is 5 cm from AB and 5 cm from BC, as shown in Figure 1.

Find the distance of the centre of mass of S from

AB,

BC.

The set square is freely suspended from C and hangs in equilibrium.

Find, to the nearest degree, the angle between CB and the vertical.

A ladder AB, of mass m and length 4 a, has one end A resting on rough horizontal ground. The other end B rests against a smooth vertical wall. A load of mass 3 m is fixed on the ladder at the point C, where AC = a. The ladder is modelled as a uniform rod in a vertical plane perpendicular to the wall and the load is modelled as a particle. The ladder rests in limiting equilibrium making an angle of 30° with the wall, as shown in Figure 2. Find the coefficient of friction between the ladder and the ground.

[ In this question, the unit vectors i and j are in a vertical plane, i being horizontal and j being vertical. ] A particle P is projected from the point A which has position vector 47.5 j metres with respect to a fixed origin O. The velocity of projection of P is (2ui + 5uj) m s^–1. The particle moves freely under gravity passing through the point B with position vector 30 i metres, as shown in Figure 3.

Show that the time taken for P to move from A to B is 5 s.

Find the value of u.

Find the speed of P at B.

A particle P of mass 2 m is moving with speed 2 u in a straight line on a smooth horizontal plane. A particle Q of mass 3m is moving with speed u in the same direction as P. The particles collide directly. The coefficient of restitution between P and Q is

Show that the speed of Q immediately after the collision is .

Find the total kinetic energy lost in the collision.

After the collision between P and Q, the particle Q collides directly with a particle R of mass m which is at rest on the plane. The coefficient of restitution between Q and R is e.

Calculate the range of values of e for which there will be a second collision between P and Q.