Edexcel A Level Further Maths Paper 3C: Further Mechanics 1 June 2024 question paper 9FM0/3C

[In this question, i and j are horizontal perpendicular unit vectors.]

A particle A has mass 3 kg and a particle B has mass 2 kg.

The particles are moving on a smooth horizontal plane when they collide directly.

Immediately before the collision, the velocity of A is (3i – j) m s^–1 and the velocity of B is (–6i + 2j) m s^–1 Immediately after the collision the velocity of A is 2i m s

Find the total kinetic energy of the two particles before the collision.

Find, in terms of i and j, the impulse exerted on A by B in the collision.

Find, in terms of i and j, the velocity of B immediately after the collision.

A rough plane is inclined to the horizontal at an angle θ, where tanθ =

A particle P of mass m is at rest at a point on the plane.

The particle moves up a line of greatest slope of the plane and comes to instantaneous rest after moving a distance d. The coefficient of friction between P and the plane is

Show that the magnitude of the frictional force acting on P as it moves up the plane

Air resistance is assumed to be negligible.

Using the work-energy principle,

find d in terms of a.

A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed of 72 km h^–1

• The resistance to the motion of the car is modelled as a constant force of magnitude 900 N

The engine of the car is working at a constant rate of PkW.

Using the model,

find the value of P.

The car now travels in a straight line up a road which is inclined to the horizontal at an

• In a refined model, the resistance to the motion of the car from non-gravitational forces is now modelled as a force of magnitude 20v newtons, where vm s^–1 is the speed of the car

At the instant when the engine of the car is working at a constant rate of 30 kW and the car is moving up the road at 10 m s^–1, the acceleration of the car is am s^–2

Using the refined model,

find the value of a.

Later on, when the engine of the car is again working at a constant rate of 30 kW, the car is moving up the road at a constant speed Um s^–1

Using the refined model,

find the value of U.

A particle A of mass 2m is moving in a straight line with speed 3u on a smooth horizontal plane. Particle A collides directly with a particle B of mass m which is at rest on the plane.

The coefficient of restitution between A and B is e, where e 0

Show that the speed of B immediately after the collision is 2u(1 + e).

After the collision, B hits a smooth fixed vertical wall which is perpendicular to the direction of motion of B.

Show that there will be a second collision between A and B.

The coefficient of restitution between B and the wall is

Find, in simplified form, in terms of m, u and e,

the magnitude of the impulse received by B in its collision with the wall,

the loss in kinetic energy of B due to its collision with the wall.

A light elastic string has natural length 2a and modulus of elasticity 2mg. One end of the string is attached to a fixed point A on a horizontal ceiling. The other end is attached to a particle P of mass m.

The particle P hangs in equilibrium at the point E, where AE = 3a. The particle P is then projected vertically downwards from E with speed

Air resistance is assumed to be negligible.

Find the elastic energy stored in the string, when P first comes to instantaneous rest. Give your answer in the form kmga, where k is a constant to be found.

[In this question, i and j are horizontal perpendicular unit vectors.]

A particle P is moving with velocity (4i – j) m s^–1 on a smooth horizontal plane. The particle collides with a smooth vertical wall and rebounds with velocity (i + 3j) m s^–1

The coefficient of restitution between P and the wall is e.

Find the value of e.

After the collision, P goes on to hit a second smooth vertical wall, which is parallel to i. The coefficient of restitution between P and this second wall is

The angle through which the direction of motion of P has been deflected by its collision with this second wall is α °.

Find the value of α, giving your answer to the nearest whole number.

A smooth uniform sphere A of mass m is moving with speed U on a smooth horizontal plane. The sphere A collides obliquely with a smooth uniform sphere B of mass 3 m which is at rest on the plane. The two spheres have the same radius.

Immediately before the collision, the direction of motion of A makes an angle α, where 0° < α < 90°, with the line joining the centres of the spheres.

Immediately after the collision, the direction of motion of A is perpendicular to its original direction, as shown in Figure 1.

The coefficient of restitution between the spheres is e.

Show that the speed of B immediately after the collision is