State the principle of conservation of momentum.

With no external force, the total momentum before a collision equals the total momentum after it.

Collision equation for two particles?

m1\mathbf{u}1+m2\mathbf{u}2=m1\mathbf{v}1+m2\mathbf{v}2

Recoil speed of a gun of mass M firing a shell of mass m at speed v (relative to the ground)?

\frac{mv}{M} in the opposite direction.

What happens to the sign of velocity in 1D if a particle moves the opposite way?

It is negative, so the momentum is negative.

Is kinetic energy conserved in a collision?

Not usually. KE after is never more than KE before (unless energy is added).

How do you conserve momentum for vector velocities?

As a vector equation: \mathbf{i} components balance, and \mathbf{j} components balance.

Why are the impulses on two colliding particles equal and opposite?

Newton's third law: the forces are equal and opposite and act for the same time.

Conservation of momentumAQA A-Level Further Maths: Flashcards

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Definition of momentum?

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Definition of momentum?: $m\mathbf{v}$ : mass times velocity, a vector in kg m s⁻¹.
State the principle of conservation of momentum.: With no external force, the total momentum before a collision equals the total momentum after it.
Collision equation for two particles?: $m_1\mathbf{u}_1+m_2\mathbf{u}_2=m_1\mathbf{v}_1+m_2\mathbf{v}_2$
Equation when two particles coalesce?: $m_1u_1+m_2u_2=(m_1+m_2)v$
What is the total momentum when a gun and shell are initially at rest?: Zero, before and after firing.
Recoil speed of a gun of mass $M$ firing a shell of mass $m$ at speed $v$ (relative to the ground)?: $\frac{mv}{M}$ in the opposite direction.
What happens to the sign of velocity in 1D if a particle moves the opposite way?: It is negative, so the momentum is negative.
Is kinetic energy conserved in a collision?: Not usually. KE after is never more than KE before (unless energy is added).
Kinetic energy lost in a collision?: KE before $-$ KE after.
How do you conserve momentum for vector velocities?: As a vector equation: $\mathbf{i}$ components balance, and $\mathbf{j}$ components balance.
Speed from the velocity $a\mathbf{i}+b\mathbf{j}$ ?: $\sqrt{a^2+b^2}$
Total momentum of $2$ kg at $(4\mathbf{i}+\mathbf{j})$ and $3$ kg at $(-2\mathbf{i}+3\mathbf{j})$ ?: $2\mathbf{i}+11\mathbf{j}$
Why are the impulses on two colliding particles equal and opposite?: Newton's third law: the forces are equal and opposite and act for the same time.

Exam questions on Conservation of momentum

Particle $A$ of mass $3$ kg, moving at $4$ m s⁻¹, collides directly with particle $B$ of mass $2$ kg, which is moving in the same direction at $1$ m s⁻¹ on a smooth horizontal surface. The particles coalesce.
Find the kinetic energy lost in the collision.2 marks
A gun of mass $800$ kg, free to recoil on smooth horizontal ground, fires a shell of mass $8$ kg horizontally. The gun and shell are initially at rest, and the shell leaves the gun with speed $250$ m s⁻¹ relative to the ground.
Find the speed of the shell relative to the gun.2 marks
Two smooth spheres $P$ and $Q$ , of masses $2m$ and $m$ , move towards each other along a straight line on a smooth horizontal table, with speeds $3u$ and $2u$ respectively. After the collision $Q$ moves in the direction of $P$ 's original motion, with speed $2u$ .
Find the velocity of $P$ after the collision.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).