Name: _______________________________________________
Edexcel A Level Further Mathematics : Further Mechanics
Work, energy and power
Date:
Time:
Total marks available:
Total marks achieved: ______
Questions
,Q1.
Two particles, A and B, are moving in opposite directions along the same straight line on a
smooth horizontal surface when they collide directly.
Particle A has mass 5m and particle B has mass 3m.
The coefficient of restitution between A and B is e, where e > 0
Immediately after the collision the speed of A is v and the speed of B is 2v.
Given that A and B are moving in the same direction after the collision,
(a) find the set of possible values of e.
(8)
Given also that the kinetic energy of A immediately after the collision is 16% of the kinetic
energy of A immediately before the collision,
(b) find
(i) the value of e,
(ii) the magnitude of the impulse received by A in the collision, giving your answer in terms of m
and v.
(6)
(Total for question = 14 marks)
Q2.
A particle P of mass 3m and a particle Q of mass 2m are moving along the same straight
line on a smooth horizontal plane. The particles are moving in opposite directions
towards each other and collide directly.
Immediately before the collision the speed of P is u and the speed of Q is 2u.
Immediately after the collision P and Q are moving in opposite directions.
The coefficient of restitution between P and Q is e.
(a) Find the range of possible values of e, justifying your answer.
(8)
Given that Q loses 75% of its kinetic energy as a result of the collision,
,(b) find the value of e.
(3)
(Total for question = 11 marks)
Q3.
Figure 2 represents the plan view of part of a horizontal floor, where AB and CD
represent fixed vertical walls, with AB parallel to CD.
A small ball is projected along the floor towards wall AB. Immediately before hitting wall
AB, the ball is moving with speed v m s–1 at an angle α to AB, where
The ball hits wall AB and then hits wall CD.
After the impact with wall CD, the ball is moving at angle
to CD.
The coefficient of restitution between the ball and wall AB is
The coefficient of restitution between the ball and wall CD is also
The floor and the walls are modelled as being smooth. The ball is modelled as a particle.
(a) Show that tan
, (7)
(b) Find the percentage of the initial kinetic energy of the ball that is lost as a result of the two
impacts.
(4)
(Total for question = 11 marks)
Q4.
A particle P, of mass m, is attached to one end of a light elastic spring of natural length a
and modulus of elasticity kmg.
The other end of the spring is attached to a fixed point O on a ceiling.
The point A is vertically below O such that OA = 3a
The pointB is vertically below O such that OB =
a
The particle is held at rest at A, then released and first comes to instantaneous rest at the point
B.
(a) Show that k =
(3)
(b) Find, in terms of g, the acceleration of P immediately after it is released from rest at A.
(3)
(c) Find, in terms of g and a, the maximum speed attained by P as it moves from A to B.
(6)
(Total for question = 12 marks)
Edexcel A Level Further Mathematics : Further Mechanics
Work, energy and power
Date:
Time:
Total marks available:
Total marks achieved: ______
Questions
,Q1.
Two particles, A and B, are moving in opposite directions along the same straight line on a
smooth horizontal surface when they collide directly.
Particle A has mass 5m and particle B has mass 3m.
The coefficient of restitution between A and B is e, where e > 0
Immediately after the collision the speed of A is v and the speed of B is 2v.
Given that A and B are moving in the same direction after the collision,
(a) find the set of possible values of e.
(8)
Given also that the kinetic energy of A immediately after the collision is 16% of the kinetic
energy of A immediately before the collision,
(b) find
(i) the value of e,
(ii) the magnitude of the impulse received by A in the collision, giving your answer in terms of m
and v.
(6)
(Total for question = 14 marks)
Q2.
A particle P of mass 3m and a particle Q of mass 2m are moving along the same straight
line on a smooth horizontal plane. The particles are moving in opposite directions
towards each other and collide directly.
Immediately before the collision the speed of P is u and the speed of Q is 2u.
Immediately after the collision P and Q are moving in opposite directions.
The coefficient of restitution between P and Q is e.
(a) Find the range of possible values of e, justifying your answer.
(8)
Given that Q loses 75% of its kinetic energy as a result of the collision,
,(b) find the value of e.
(3)
(Total for question = 11 marks)
Q3.
Figure 2 represents the plan view of part of a horizontal floor, where AB and CD
represent fixed vertical walls, with AB parallel to CD.
A small ball is projected along the floor towards wall AB. Immediately before hitting wall
AB, the ball is moving with speed v m s–1 at an angle α to AB, where
The ball hits wall AB and then hits wall CD.
After the impact with wall CD, the ball is moving at angle
to CD.
The coefficient of restitution between the ball and wall AB is
The coefficient of restitution between the ball and wall CD is also
The floor and the walls are modelled as being smooth. The ball is modelled as a particle.
(a) Show that tan
, (7)
(b) Find the percentage of the initial kinetic energy of the ball that is lost as a result of the two
impacts.
(4)
(Total for question = 11 marks)
Q4.
A particle P, of mass m, is attached to one end of a light elastic spring of natural length a
and modulus of elasticity kmg.
The other end of the spring is attached to a fixed point O on a ceiling.
The point A is vertically below O such that OA = 3a
The pointB is vertically below O such that OB =
a
The particle is held at rest at A, then released and first comes to instantaneous rest at the point
B.
(a) Show that k =
(3)
(b) Find, in terms of g, the acceleration of P immediately after it is released from rest at A.
(3)
(c) Find, in terms of g and a, the maximum speed attained by P as it moves from A to B.
(6)
(Total for question = 12 marks)
