Edexcel A Level Further Maths Paper 3C: Further Mechanics 1 June 2025 question paper 9FM0/3C

A particle A has mass 4m and a particle B has mass 3m. The particles are moving along the same straight line on a smooth horizontal table. The particles are moving in opposite directions towards each other when they collide directly.

As a result of the collision, the direction of motion of each particle is reversed.

Immediately before the collision, the speed of A is u and the speed of B is ku, where k is a constant.

Immediately after the collision, the speed of A is 2v and the speed of B is 3v.

The magnitude of the impulse received by A in the collision is 20mv.

Find u in terms of v only.

Find the exact value of k.

A van of mass 900kg is moving along a straight horizontal road.

When the speed of the van is v ms¹, the resistance to the motion of the van is modelled as a force of magnitude 50 v N.

The engine of the van is working at a constant rate of 12 kW.

At the instant when the speed of the van is V ms¹, the acceleration of the van is 0 2. ms²

Using the model,

find the value of V.

Later on, the van is moving up a straight road that is inclined at an angle α to the horizontal, where sin α When the speed of the van is v ms¹, the resistance to the motion of the van from non-gravitational forces is modelled as a force of magnitude 50 v N.

When the van has constant speed U ms⁻¹, the engine of the van is working at a constant rate of 15kW.

Using the model,

find the value of U.

A particle P of mass 0.4 k moving with velocity 7i ms¹ when it receives

Find the two possible values of λ

A rough straight ramp is fixed to horizontal ground. The ramp is inclined at an angle θ to the horizontal, where sin θ The points A and B are on a line of greatest slope of the ramp with AB 18 m and B above A, as shown in Figure 1.

A package of mass 0.5 kg is projected up the ramp from A with speed 8 m s¹ and comes to instantaneous rest at B.

The work done against friction as the package moves from A to B is W joules.

The package is modelled as a particle and air resistance is ignored.

Use the work-energy principle to show that W = 9.7

The coefficient of friction between the package and the ramp is μ

Find the value of μ

Three small balls, A, B and C, have masses 2m, 3m and 4m respectively.

The balls are initially at rest in a straight line on a smooth horizontal surface, with B between A and C, as shown in Figure 2.

Ball A is projected towards B with speed 5u and A and B collide directly. Immediately after the collision, the speed of B is 3 u

The balls are modelled as uniform spheres with equal radii.

Find, in terms of u, the speed of A immediately after the collision with B.

Find the coefficient of restitution between A and B.

After the collision between A and B, ball C is projected towards B with speed u Balls B and C collide directly. The coefficient of restitution between B and C is f

Given that there is a second direct collision between A and B

find the complete range of possible values of f

A smooth plane is inclined at an angle θ to horizontal ground, where sin θ The points A, B and C lie on a line of greatest slope of the plane, with B between A and C, with A above B and AC = 3a, as shown in Figure 3.

A light elastic string of natural length 2a has one end attached to the fixed point A. The other end of the string is attached to a parcel P of mass m. The modulus of elasticity of the string is mg The parcel rests in equilibrium on the plane at the point B.

The parcel is modelled as a particle.

The parcel is now held at C and released from rest.

Find the elastic potential energy lost by the string when P moves from C to B.

Use the principle of conservation of mechanical energy to find the speed of P at the instant it passes B.

Two uniform spheres P and Q have equal radii. The mass of P is 0.3 kg and the mass of Q is 0.4 kg. The spheres are moving on a smooth horizontal plane when they collide obliquely.

Immediately before the collision,

• P is moving with speed 5 ms¹ at 30° to the line of centres of the spheres

• Q is moving with speed 3 ms¹ at 60° to the line of centres of the spheres

Immediately after the collision,

• P is moving with speed v ms¹ at 60° to the line of centres of the spheres

• Q is moving with speed w ms¹ at θ to the line of centres of the spheres

as in the plan view shown in Figure 4.

Find (i) the value of w

(ii) the value of θ

Find the coefficient of restitution between P and Q.

Figure 5 represents the plan view of part of a smooth horizontal floor, where RS and ST are smooth fixed vertical walls. The vector RS→ is in the direction of the vector i. The vector ST→ is in the direction of the vector 3i 4j.

A small ball B of mass 0 25. kg is projected across the floor towards RS.

Immediately before the impact with RS, the velocity of B is v ms¹ Immediately after the impact with RS, the velocity of B is 3i 2j ms₁ The coefficient of restitution between B and RS is The ball is modelled as a particle.

Show that the kinetic energy lost by B in the impact with RS is 4 J.

Immediately after the impact with ST, the velocity of B is w ms¹ Given that the coefficient of restitution between B and ST is

find w in terms of i and j.