Name: _______________________________________________
Edexcel A Level Further Mathematics : Further Mechanics
Elastic collisions in one dimension
Date:
Time:
Total marks available:
Total marks achieved: ______
Questions
,Q1.
Figure 1 represents the plan of part of a smooth horizontal floor, where W1 and W2 are
two fixed parallel vertical walls. The walls are 3 metres apart.
A particle lies at rest at a point O on the floor between the two walls, where the point O
is d metres, 0 < d ≤ 3, from W1
At time t = 0, the particle is projected from O towards W1 with speed u m s–1 in a direction
perpendicular to the walls.
The coefficient of restitution between the particle and each wall is
The particle returns to O at time t = T seconds, having bounced off each wall once.
(a) Show that T =
(6)
The value of u is fixed, the particle still hits each wall once but the value of d can now vary.
(b) Find the least possible value of T, giving your answer in terms of u. You must give a reason
for your answer.
(2)
(Total for question = 8 marks)
Q2.
Two particles, P and Q, are moving in opposite directions along the same straight line on a
smooth horizontal surface when they collide directly.
The mass of P is 3m and the mass of Q is 4m.
Immediately before the collision the speed of P is 2u and the speed of Q is u.
The coefficient of restitution between P and Q is e.
, (a) Show that the speed of Q immediately after the collision is u e
(9e + 2)
(6)
After the collision with P, particle Q collides directly with a fixed vertical wall and rebounds. The
wall is perpendicular to the direction of motion of Q.
The coefficient of restitution between Q and the wall is
(b) Find the complete range of possible values of e for which there is a second collision between
P and Q.
(4)
(Total for question = 10 marks)
Q3.
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)
Edexcel A Level Further Mathematics : Further Mechanics
Elastic collisions in one dimension
Date:
Time:
Total marks available:
Total marks achieved: ______
Questions
,Q1.
Figure 1 represents the plan of part of a smooth horizontal floor, where W1 and W2 are
two fixed parallel vertical walls. The walls are 3 metres apart.
A particle lies at rest at a point O on the floor between the two walls, where the point O
is d metres, 0 < d ≤ 3, from W1
At time t = 0, the particle is projected from O towards W1 with speed u m s–1 in a direction
perpendicular to the walls.
The coefficient of restitution between the particle and each wall is
The particle returns to O at time t = T seconds, having bounced off each wall once.
(a) Show that T =
(6)
The value of u is fixed, the particle still hits each wall once but the value of d can now vary.
(b) Find the least possible value of T, giving your answer in terms of u. You must give a reason
for your answer.
(2)
(Total for question = 8 marks)
Q2.
Two particles, P and Q, are moving in opposite directions along the same straight line on a
smooth horizontal surface when they collide directly.
The mass of P is 3m and the mass of Q is 4m.
Immediately before the collision the speed of P is 2u and the speed of Q is u.
The coefficient of restitution between P and Q is e.
, (a) Show that the speed of Q immediately after the collision is u e
(9e + 2)
(6)
After the collision with P, particle Q collides directly with a fixed vertical wall and rebounds. The
wall is perpendicular to the direction of motion of Q.
The coefficient of restitution between Q and the wall is
(b) Find the complete range of possible values of e for which there is a second collision between
P and Q.
(4)
(Total for question = 10 marks)
Q3.
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)
