Pearson Edexcel IAL Mathematics M2.4.1 Momentum as a vector

Practise using vector momentum and impulse to find velocities, speeds, angles and impulse magnitudes for particles before and after impact.

Exam points

resolve impulse-momentum equations in i and j to find a velocity vector after impact
convert velocity components into speed or direction angles using magnitude and trigonometry
use conservation of linear momentum with impulse changes to solve for unknown speeds

[Maximum number: 8]

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

M1

8mu-6mu=2my-4mx

A1

Equation for kinetic energy

M1

$2mx^2+my^2=\frac12(2m\cdot4u^2+m\cdot9u^2)$

A1

$17u^2=2y^2+(y-u)^2 \Rightarrow 3y^2-2uy-16u^2=0$

DM1

$y=\frac83u$

A1*

(b)

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

$I=\pm2m\left(y-\left(-\frac14y\right)\right)$

M1

$|I|=\frac{20}{3}mu$

A1

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M2.4.1 - Momentum as a vector question 1

Question (a)

Question (b)