Edexcel A-Level Mathematics A2 M2.4 Collisions Questions

Practise M2 collisions by combining momentum, impulse, restitution, kinetic energy loss and wall rebounds in one- or two-particle models.

Exam points

Pair conservation of momentum with restitution to find post-collision speeds or ranges for e.
Use impulse as change in momentum, including vector components and wall rebounds.
Check second-collision conditions by comparing rebound speeds after applying f or e.

Covered syllabus points

M2.4.1 - Momentum as a vector M2.4.2 - Direct impact of elastic particles M2.4.3 - Successive impacts of up to three

[Maximum number: 14]

Particle P has mass 4 m and particle Q has mass 2 m.

The particles are moving in opposite directions along the same straight line on a smooth horizontal surface.

Particle P collides directly with particle Q.
Immediately before the collision, the speed of P is 2 u and the speed of Q is 3 u.
Immediately after the collision, the speed of P is x and the speed of Q is y.
The direction of motion of each particle is reversed as a result of the collision.
The total kinetic energy of P and Q after the collision is half of the total kinetic energy of P and Q before the collision.

(a)

Show that $y=\frac{8}{3} u$

The coefficient of restitution between P and Q is e.

[ 6 ]

Official answer figure

Equation for CLM
8mu-6mu=2my-4mx
Equation for kinetic energy
$2mx^2+my^2=\frac12(2m\cdot4u^2+m\cdot9u^2)$ $17u^2=2y^2+(y-u)^2 \Rightarrow 3y^2-2uy-16u^2=0$ $y=\frac83u$

A1*

(b)

Find the value of e.

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

Particle Q rebounds.
The coefficient of restitution between Q and the wall is f.
Given that there is no second collision between P and Q,

[ 3 ]

Use of impact law:

x+y=5eu $e=\frac{\frac12\left(\frac83u-u\right)+\frac83u}{5u}$ $e=\frac{7}{10}$

(c)

find the range of possible values of f.

Given that $f=\frac{1}{4}$

[ 3 ]

Velocity of Q after impact with the wall:

$f\cdot\frac83u$
No second collision if

$f\cdot\frac83u\le \frac56u$ $0<f\le\frac{5}{16}$

(d)

find, in terms of m and u, the magnitude of the impulse received by Q as a result of its impact with the wall.

[ 2 ]

Use

$I=\pm2m\left(y-\left(-\frac14y\right)\right)$ $|I|=\frac{20}{3}mu$

[Maximum number: 14]

Particle P has mass m and particle Q has mass 5 m.

The particles are moving in the same direction along the same straight line on a smooth horizontal surface.

Particle P collides directly with particle Q.
Immediately before the collision, the speed of P is 6 u and the speed of Q is u.
Immediately after the collision, the speed of P is x and the speed of Q is y.
The direction of motion of P is reversed as a result of the collision.
The coefficient of restitution between P and Q is e.

(a)

Find the complete range of possible values of e.

[ 7 ]

Figure for Answer to Question (a) — Edexcel A-Level Mathematics A2

6mu+5mu=5my-mx\Rightarrow 11u=5y-x

x+y=5eu

6x=25eu-11u,\qquad x>0\Rightarrow 25e>11

\frac{11}{25}<e<1

(b)

Given that $e=\frac35$ , find the total kinetic energy lost in the collision between P and Q.

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

Particle Q rebounds.

The coefficient of restitution between Q and the wall is f.

Given that there is a second collision between P and Q,

[ 4 ]

x=\frac23u,\qquad y=\frac73u

M1 A1ft

\text{Total KE lost}=\left(\frac12m\cdot 36u^2+\frac12\cdot 5m\cdot u^2\right)-\left(\frac12m x^2+\frac12\cdot 5m y^2\right)

=\left(\frac12m\cdot 36u^2+\frac12\cdot 5m\cdot u^2\right)-\left(\frac12m\cdot \frac49u^2+\frac12\cdot 5m\cdot \frac{49}{9}u^2\right)

=\frac{20}{3}mu^2

(c)

find the complete range of possible values of f.

[ 3 ]

velocity of Q after collision with wall
$= \pm f y \quad\left(= \pm f \times \frac{7}{3} u\right)$

B1ft

Follow their y (in terms of u )

Second collision if $f y>x \frac{7}{3} f u>\frac{2}{3} u$
Correct inequality for their x, y
Dependent on the B1 and P moving
away from the wall

$\frac{2}{7}<f,, 1$
Correct only
Need both limits

[3]

(14)

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question (d)

Question 2

Question (a)

Question (b)

Question (c)