Edexcel A Level Further Maths Paper 3C: Further Mechanics 1 June 2023 question paper 9FM0/3C

A particle P of mass 2 kg is moving with velocity (4i 3j) m s¹when it receives an impulse (6i 42j) N s.

Find the speed of P immediately after receiving the impulse.

The angle through which the direction of motion of P has been deflected by the impulse is α °

Find the value of α

A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed U m s⁻¹. The resistance to the motion of the car is a constant force of magnitude 400 N.

The engine of the car is working at a constant rate of 16 kW.

Find the value of U.

The car now pulls a trailer of mass 600 kg in a straight line along the road using a tow rope which is parallel to the direction of motion. The resistance to the motion of the car is again a constant force of magnitude 400 N. The resistance to the motion of the trailer is a constant force of magnitude 300 N.

The engine of the car is working at a constant rate of 16 kW.

The tow rope is modelled as being light and inextensible.

Using the model,

A particle P of mass 2 m is moving in a straight line with speed 3 u on a smooth horizontal plane. It collides directly with a particle Q of mass m that is moving on the plane with speed 2 u in the opposite direction to P.

The coefficient of restitution between P and Q is e, where e >

After the collision Q hits a smooth fixed vertical wall that is perpendicular to the direction of motion of Q. The coefficient of restitution between Q and the wall is f.

Find, in terms of e, the set of values of f for which there will be a second collision between P and Q.

A light elastic string has natural length 2 a and modulus of elasticity 4 mg. One end of the elastic string is attached to a fixed point O. A particle P of mass m is attached to the other end of the elastic string. The particle P hangs freely in equilibrium at the point E, which is vertically below O

Find the length OE.

Particle P is now pulled vertically downwards to the point A, where OA = 4a, and released from rest. The resistance to the motion of P is a constant force of

Find, in terms of a and g, the speed of P after it has moved a distance a.

Particle P is now held at O Particle P is released from rest and reaches its maximum speed at the point B. The resistance to the motion of P is again a constant force of magnitude mg.

Find the distance OB.

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

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

Immediately after the collision the direction of motion of S makes an angle β with the line joining the centres of the spheres, as shown in Figure 1.

The coefficient of restitution between the spheres is e.

Given that m = eM,

show that the directions of motion of the two spheres immediately after the collision are perpendicular.

A particle P of mass m is falling vertically when it strikes a fixed smooth inclined plane. The plane is inclined to the horizontal at an angle α, where 0 < 45 At the instant immediately before the impact, the speed of P is u.

At the instant immediately after the impact, P is moving horizontally with speed v.

Show that the magnitude of the impulse exerted on the plane by P is mu sec

The coefficient of restitution between P and the plane is e, where e > 0

Show that v² u²(sin² e² cos²)

Show that the kinetic energy lost by P in the impact is

Hence find, in terms of m, u and e only, the kinetic energy lost by P in the impact.

A small smooth snooker ball is projected from the corner A of a horizontal rectangular snooker table ABCD.

The ball is projected so it first hits the side DC at the point P, then hits the side CB at the point Q and then returns to A.

Angle APD = α, Angle QPC = β, Angle AQB = γ The ball moves along AP with speed U, along PQ with speed V and along QA with speed W, as shown in Figure 2.

The coefficient of restitution between the ball and side DC is e₁

The coefficient of restitution between the ball and side CB is e₂

The ball is modelled as a particle.

Use the model to answer all parts of this question.

Show that tan e₁ tan

Hence show that e₁ tan e₂ cot

By considering (angle APQ + angle AQP) or otherwise, show that it would be possible for the ball to return to A only if e₂ > e₁

If instead e₁=e₂, the ball would not return to A.

Given that e₁=e₂

use the result from part (b) to describe the path of the ball after it hits CB at Q, explaining your answer.