State the impulse-momentum principle.

\mathbf{I}=m\mathbf{v}-m\mathbf{u}: impulse equals change in momentum.

How do you find the velocity after an impulse?

\mathbf{v}=\mathbf{u}+\frac{\mathbf{I}}{m}.

State conservation of linear momentum.

Total momentum before = total momentum after, if no external impulse acts.

Momentum equation for a collision with coalescence?

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

How are the impulses on two colliding particles related?

Equal in magnitude and opposite in direction (Newton's third law).

No: speed is the magnitude of the velocity vector.

How do you split a vector momentum equation?

Into an \mathbf{i}-equation and a \mathbf{j}-equation, solved separately.

How do you find the angle turned through by a ball?

Subtract the direction angles of \mathbf{u} and \mathbf{v} (check quadrants), or use the scalar product.

Momentum and impulse in vector formEdexcel International A Level Further Maths: Flashcards

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What is the momentum of a particle of mass $m$ and velocity $\mathbf{v}$?

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What is the momentum of a particle of mass $m$ and velocity $\mathbf{v}$ ?: $m\mathbf{v}$ , a vector in the direction of $\mathbf{v}$ .
Units of impulse?: N s (the same as kg m s $^{-1}$ ).
Impulse in terms of force and time?: $\mathbf{I}=\mathbf{F}t$ .
State the impulse-momentum principle.: $\mathbf{I}=m\mathbf{v}-m\mathbf{u}$ : impulse equals change in momentum.
How do you find the velocity after an impulse?: $\mathbf{v}=\mathbf{u}+\frac{\mathbf{I}}{m}$ .
Magnitude of $a\mathbf{i}+b\mathbf{j}$ ?: $\sqrt{a^2+b^2}$
State conservation of linear momentum.: Total momentum before $=$ total momentum after, if no external impulse acts.
Momentum equation for a collision with coalescence?: $m_1\mathbf{u}_1+m_2\mathbf{u}_2=(m_1+m_2)\mathbf{v}$ .
How are the impulses on two colliding particles related?: Equal in magnitude and opposite in direction (Newton's third law).
Is speed a vector?: No: speed is the magnitude of the velocity vector.
How do you split a vector momentum equation?: Into an $\mathbf{i}$ -equation and a $\mathbf{j}$ -equation, solved separately.
How do you find the angle turned through by a ball?: Subtract the direction angles of $\mathbf{u}$ and $\mathbf{v}$ (check quadrants), or use the scalar product.

Exam questions on Momentum and impulse in vector form

A ball of mass $0.4$ kg is moving with velocity $(6\mathbf{i}+8\mathbf{j})$ m s $^{-1}$ when it is hit by a bat. Immediately after being hit it moves with velocity $(-2\mathbf{i}+5\mathbf{j})$ m s $^{-1}$ . The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are perpendicular and lie in a horizontal plane.
Find the angle between the direction of the impulse and the vector $-\mathbf{i}$ .2 marks
Two particles $A$ and $B$ , of masses $2$ kg and $3$ kg, move on a smooth horizontal plane and collide. Before the collision $A$ has velocity $(4\mathbf{i}-\mathbf{j})$ m s $^{-1}$ and $B$ has velocity $(-2\mathbf{i}+3\mathbf{j})$ m s $^{-1}$ . After the collision $A$ has velocity $(-\mathbf{i}+2\mathbf{j})$ m s $^{-1}$ . The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are perpendicular and lie in the plane.
Find the speed of $B$ after the collision.2 marks
A tennis ball of mass $0.06$ kg is moving with velocity $(-20\mathbf{i}+5\mathbf{j})$ m s $^{-1}$ when it is struck by a racket. The racket exerts an impulse of $(3\mathbf{i}+1.5\mathbf{j})$ N s on the ball. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are perpendicular and lie in a horizontal plane. Ignore the weight of the ball.
Find the velocity of the ball immediately after it is struck.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).